JOURNAL OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 4, Number I, January 1991
OSCILLATION THEOREMS
FOR PRIMES IN ARITHMETIC PROGRESSIONS
AND FOR SIFTING FUNCTIONS
JOHN FRIEDLANDER, ANDREW GRANVILLE,
ADOLF HILDEBRAND, AND HELMUT MAIER
CONTENTS
I. Introduction
1. Primes in arithmetic progressions
2. The main theorems
II. Oscillation theorems for sifting functions
3. The statements
4. Proof of Theorem B2
5. Proof of Proposition 3.1
6. Proof of Proposition 3.2
7. Proof of Theorem C: preliminaries
8. Proof of Theorem C: completion
9. Proof of Theorem Bl
10. Proof of Theorem B3
III. Oscillation theorems for primes
11. Proof of Theorem Al
12. Proof of Theorem A2
13. Proofs of Theorem A3 and Proposition 2.1
I.
1.
INTRODUCTION
PRIMES IN ARITHMETIC PROGRESSIONS
For x ~ 2 real, q a positive integer, and a an integer coprime to q, let 0
and ~ be defined by
(1.1)
O(x; q, a)
=
E
logp
= qJZq) (1 +~(x; q, a)),
p~x
p=amodq
Received by the editors June 15, 1989.
1980 Mathematics Subject Classification (1985 Revision). Primary 11N13; Secondary llN25,
11N35.
The first two authors were partially supported by NSERC and the last two authors were partially
supported by NSF.
© 1991
American Mathematical Society
0894-0347/91 $1.00 + $.25 per page
25
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26
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
where rp is Euler's function. The prime number theorem for arithmetic progressions is equivalent to the statement that .1o(x; q, a) = 0(1) holds, as x -+ 00,
for fixed q and a. The function .10 thus provides a measure for the regularity
of distribution of primes in arithmetic progressions.
The main focus in the study of primes in arithmetic progressions has previously been on estimating .1o(x; q, a) from above uniformly for all, or almost
all, q :5 Q, with Q as large as possible. The two fundamental theorems in
this connection (cf. [Da)) are the Siegel-Walfisz and the Bombieri-Vinogradov
theorems. It follows from the former that the bound:
(1.2)
1.1o(x; q, a)1
«
(logx)
-A
holds uniformly for q :5 (logx)B , for any fixed positive constants A and B.
The latter states, in one form, that
( 1.3)
"L..J -(-)
1 max 1.1o(x; q, a)1
<Q rp q (a,q)=1
q-
«
(logx) -A
holds for any A > 0 with Q = y'X(logx)-B, where B is a suitable constant
depending on A. As a consequence, the estimate (1.2) is true for all but o(Q)
moduli q :5 Q, provided that Q :5 y'X(logx)-c with a sufficiently large constant C.
It is generally expected that these results are far from being the best possible.
In fact, under the Generalized Riemann Hypothesis, (1.2) is known to hold for
all q :5 y'X(logx)-B and all a coprime with q, if B is a sufficiently large
constant. However, even this conditional result probably is not optimal, and
stronger conjectures about the size of .1o(x; q, a) and the range of validity of
estimates like (1.2) or (1.3) have been put forward.
Perhaps the best known of these conjectures is the Elliott-Halberstam conjecture [EH], according to which the Bombieri-Vinogradov estimate is valid for
Q = x 1- e and possibly even for Q = x(logX)-B with a sufficiently large constant B; see [EH, p. 62] for the precise formulation. This conjecture was at
least partially motivated by probabilistic considerations; in fact, Elliott [E1]
had shown earlier that, in an appropriate sense, "almost all" primelike sequences of positive integers satisfy a Bombieri-Vinogradov type estimate (1.3)
with Q = x(logx)-B, B = B(A).
A corresponding conjecture for the estimate of individual terms .1o(x; q, a)
has been made by Montgomery [Mo]. It implies that
( 1.4)
1.1o(x; q, a)1
«e (qjx) 1/2-e logx
holds uniformly for q :5 x, for any given e > O. A weaker form of (1.4) may
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OSCILLATION THEOREMS
27
be given as
( 1.5)
The latter estimate is only of interest for q ~ x 1- e , while (1.4) remains nontrivial for moduli as large as x divided by a fixed power of log x .
The strongest forms of both of these conjectures were recently disproved by
Friedlander and Granville [FG]. They showed that for any A > 0 there exist
arbitrarily large values of x and integers q ~ x(logx)-A and (a, q) = 1,
for which I.:\(x; q, a)1 » 1, thus disproving Montgomery's conjecture (1.4).
Moreover, such lower bounds hold for most moduli q, so that (1.2) even fails
when averaged over a suitable range for q. As a consequence, the BombieriVinogradov type estimate (1.3) cannot be valid with Q as large as x divided
by any fixed power of log x .
The main purpose of this paper is to extend these results by showing that
the mentioned conjectures already fail for moduli that are significantly smaller
than x divided by any fixed power of log x . Specifically, we shall prove that
(1.4) fails to hold for all moduli q as small as x exp(-(logx) 1/5-0) , most
moduli q as small as xexp(-(logx)I/3-0), and many moduli q as small as
x exp( - (log x) 1/2-0), for any fixed 0 > 0, if the parameter e in (1.4) is sufficiently small. We also show that, given any positive constant A, the estimate
(1.3) is false for all sufficiently large x with Q=x exp{ -(1-e)A(log2 x)2/log3x} ,
where throughout, logk x denotes the k times iterated logarithm.
We state our main results in three theorems, Theorems A1-A3, which are
proven by rather different methods. The first method is closest in spirit to the
argument of Friedlander and Granville [FG]. It has its roots in a paper of Maier
[Ma], in which he showed that the asymptotic formula n(x+y)-n(x) '" y/ logx
fails to hold when y = (logx)A ,for any fixed positive;". The second method is a
modification of the first, while the third is new and quite different from the other
two approaches. These ideas are briefly discussed following the statements.
Common to all three methods is the fact that they ultimately depend on
oscillation results for certain sifting functions, a typical example being the wellknown function:
<l>(y, z)
= #{n ~ y: pin => p ~ z}.
Such oscillation results turn out to be quite difficult to prove. While it is relatively easy to obtain upper bounds for the oscillations of functions like <l>(y, z) ,
thereby showing that these functions behave in a regular manner to some extent, it is much harder to exhibit irregularities of such functions. In Chapter II
we prove three such oscillation theorems, Theorems B1-B3, which are applied
to prove Theorems A 1-A3, respectively in Chapter III. Two of these theorems
depend on a further and more fundamental result, Theorem C, which gives an
"asymptotic formula" for the amount of oscillations of the function <l>(y, z) .
This theorem is quite difficult to prove and takes up the bulk of Chapter II.
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28
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
2.
THE MAIN THEOREMS
We consider here only the error function Ll( x; q , a) defined in (1.1). It is
perhaps more customary to deal with the functions Li, ~ defined by
n(x; q, a) =
I:
lix
p5,x
p=amodq
and
x
If/(x; q, a) =
n<x
n=amodq
~
1 = q1(q)(1 +Ll(x; q, a))
A(n) = q1(q) (1
-
+ Ll(x;
q, a)).
In fact, although we find Ll slightly simpler to work with, similar results hold
for Li, ~.
Our first theorem gives lower bounds for the oscillations of the function
Ll(x; q, a) in the case of moduli q that do not have too many small prime
factors.
Theorem At. Let e > O. Then for all q 2: qo (e) and all x satisfying
I:
plq
(2.1 )
p<\ogq
q(logq) l+e < x:::; qexp (( logq )5/11-e ),
(2.2)
there exist numbers x± with x/2 < x± :::; 2x and integers a± coprime with q
such that
(2.3)
Ll(x'
(2.4)
where y
a) < _ -(I+e)OI(X,y)
-,q, - -
= x/q
and J 1(x, y)
y
,
= log(logy / log2 x)/ log(logx/logy).
The right-hand sides of (2.3) and (2.4) are» 1 if y = (logxf(1) , and are
»y-(I+e)a/(I-a) if y = exp((logxt) with 0 < (); :::; 1/3. Thus the theorem
shows that Montgomery's conjecture (1.4) is incorrect for moduli q as small as
xexp(_(logX)I/3-0) , for any given positive J.
The lower bounds in (2.3) and (2.4) as well as the range (2.2) for x are
essentially the limits of what can be achieved by our method of proof. In
connection with the lower bound in (2.2) it should be pointed out that for
x < (1 - e)q1(q) logq , large lower bounds for Ll follow trivially from the fact
that there are more reduced classes modulo q than there are primes :::; x. As
far as the upper bound is concerned, we note that the theorem is not of much
interest in the range q < x exp( _(logX)I/3+e), since then J 1(x, y) > 1/2 and
the lower bounds in (2.3) and (2.4) become smaller than the bounds predicted
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OSCILLATION THEOREMS
29
by conjectures like (1.4). Under the assumption of the Generalized Riemann
Hypothesis, the exponent 01 (x ,y) can be reduced to log(logy /1og2 x)/ 10g2 x .
Theorem Al sharpens and extends to a much larger range the corresponding result of Friedlander and Granville [FG], and represents the analogue for
arithmetic progressions of a recent result of Hildebrand and Maier [HM] on
irregularities in the distribution of primes in short intervals. This latter result
asserts that bounds of the type (2.3) and (2.4) hold with the same function 01
for all sufficiently large x and 2 :::; y :::; exp(A(10gx)I/3), if ~(x±; q, a±) is
replaced by ~(x±, y) for suitable x±, x:::; x± :::; 2x, where
~(x , y) =.!.
I:
y x<p:::;x+y
logp - 1
and A is a positive constant.
The proof of Theorem Al combines the methods of [FG] and [HM]. The
basic idea, which is essentially due to Maier [Ma], is to consider the occurrence
of primes in a rectangular array rP + qs, where r and s run independently
through the integers in certain intervals. Here q is the modulus of the progression in which we are interested. Thus, for each fixed r and varying s, we
are considering the primes in a progression modulo q and trying to prove that
this number is not always close to what we "expect." On the other hand, for
each fixed s and varying r, we are considering the primes in an arithmetic
progression modulo P. Here P is chosen much smaller than q so that the
number of such primes can be accurately estimated. This estimate, which rests
on the Linnik-Gallagher prime number theorem, then reduces the problem to
counting the number of s satisfying (s, P) = 1. Now, if this number is more
(or less) than the expected number, then there will be correspondingly more (or
less) primes than expected in some progression modulo q.
Thus one is led to study oscillations of sifting functions of the type
#{s :::; y: (s, P) = I}. In the case where P is of the form P = TIp<zP'
these functions reduce to the well-known function <J>(y, z). Oscillation theorems for <J>(y, z) have been given in [Ma] (see also [Iw, p. 211]) for the range
logy « logz, and in [HM] for the range logy « zl/2-e. These results would
not be sufficient for the proof of Theorem AI, however, and therefore, we prove
a new oscillation result, Theorem B 1, which is much sharper than the previous
results.
The above argument requires that (q, P) = 1 and thus, when P is chosen
as the product of all primes < z , it applies only to moduli q having no prime
factors < z. For more general q it is necessary to modify the choice of P and,
hence, also the definition of <J>(y, z) , to exclude the small prime factors of q.
Provided q does not have too many small prime divisors, the modified function
can be shown to exhibit essentially the same oscillatory behavior. This is the genesis of condition (2.1). It is possible to deduce from the Tunin-Kubilius inequality that almost all integers q have (1 + 0(1)) ~P<logq(l/p) = (1 + 0(1)) 10g3 q
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30
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
distinct prime factors less than log q and so almost all integers satisfy (2.1)
(see, e.g., [E2, Chapter 4]).
Condition (2.1) could probably be relaxed to some extent, but in this approach it cannot be completely removed without sacrificing the quality of the
result. Our next theorem gives a result that is somewhat weaker than the estimates of Theorem AI, but has the advantage of being valid for all q.
Theorem A2. Let e > O. There exist N(e) > 0 and % = qo(e) > 0 such that
for any q > qo and any x with
q(logq)
(2.5)
N(e)
1/3
<x:::;qexp((logq)),
there exist numbers x± with x/2 < x± :::; 2x and integers a± coprime with q
such that
d('
(2.6)
)
>
1
og x
x+ ' q, a+ - -1-5-y
-(I+e)02(x ,y)
,
a) < __1_ -(i+e)02(X ,y)
-,q, - - 1 5 Y
,
og x
d(X'
(2.7)
where y = x/q and C>2(X, y) = 3log(logy/log2x)/10g(10gxlogy).
This result is obtained by taking P in the above argument to be a product
of primes chosen from an appropriate short interval. One consequence of this
is that we are able to exclude all prime factors of q even if there are many of
these. Another is that the close proximity of all prime factors of P allows us
to deduce the oscillation result by combinatorial means.
The factor l/log 5 x is significant only if logy « (10g2 X)2. If two-sided
estimates are not required and only a lower bound for Idl is sought, then this
factor may be replaced by logy /(10g2 X)2 ; see the remarks after the proof.
As compared with Theorem AI, the exponent C>2 is somewhat worse than
C>I ' being less than
only for y < exp(logl/5-e x). (Assuming the Generalized
Riemann Hypothesis this too may be improved.) However, Theorem A2 has
the advantage of being valid for all q and has the additional advantage that the
required oscillation result, Theorem B2 (see §§3, 4), is much easier to prove.
Next we turn to lower bounds for averages over d(x; q, a). As almost all
q satisfy the hypothesis of Theorem AI, we immediately obtain from Theorem
Al the following result:
1
Corollary 2.1. For all sufficiently large x and 2x exp( _(logx)5/II-e) :::; Q :::;
x/2, we have
(2.8)
'"
1
,
£- - - rr,tax max Id(X ; q, a)1 »y
rp(q) x:5:.x (a,q)=1
q:5:.Q
-(I+e)o (x y)
1',
where y = x/Q.
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31
OSCILLATION THEOREMS
Theorem A2 yields a similar, but slightly weaker result. Stronger results may
be obtained by bounding from below averages of the type
L
(2.9)
q<Q
(q,a)=1
or
_(1)1L\(x; q, a)1
rp q
L
(2.10)
q"'Q
(q,a)=1
-hL\(X; q, a)
rp q
=~
=~
L
IO(X; q, a) _
L
(O(X; q, a) -
q<Q
(q,a)=1
q"'Q
(q,a)=1
x( )1
rp q
rp~q))
,
where a is a fixed nonzero integer and q '" Q means Q < q :::; 2Q. Such averages have been considered in recent work of Fouvry [Fo], and Bombieri, Friedlander, and Iwaniec [BPI], where nontrivial upper bounds have been given for
(2.9) and (2.10) in ranges beyond the known range of validity of the BombieriVinogradov theorem in its original form (1.3). Thus, it is of interest to obtain
lower bounds for these quantities similar to the bound (2.8), and, of course,
such bounds provide, a fortiori, a lower bound for the left-hand side of (2.8).
Our third theorem gives such a result.
Theorem A3. Let e > O. Then for some c > 0, for all x> xo(e) , and all Q
satisfying
x exp( -cVlogx) < Q :::; x(10gx)-(1+e) ,
(2.11 )
there exist Q± with Q/2 < Q± :::; 2Q and integers a± such that
rp ( q )
"
(2.13)
L..J
q"'Q_
(q,a_)=1
where y
a) > _1_ -(1+e)o3(x ,y)
, q, + - 10g2 X Y
,
_l_L\(X'
(2.12)
= x/Q
_l_L\(x· q a ) < __1_ y -(1+e)03(X,y)
rp (q )
"- log x
'
2
and 03 (X ,y) = 10g(10gy/log2 x)110g2 X.
We remark that 03 is smaller, and hence better, than the exponents 01 and
02 from Theorems Al and A2, and indeed is the same exponent achieved by
the method of Theorem A 1 on the assumption of the Generalized Riemann Hypothesis. The result remains of interest in the entire range y< exp( (log x) 1/ 2-e)
and, with more effort, could be somewhat sharpened so that the bound remained
larger than y-1 / 2 in the range y < exp(cvlogx) .
The factor 1/log2 x in (2.12) and (2.13) is significant only for logy «
10g2 x 10g3 x. In fact, for this latter range it can also be shown that this factor
may be dropped, if one uses the method that gives Theorem AI, provided that
the sum E q"'Q,(q,a±)=1 is replaced by the sum Eq",Q max(a,q)=1 (respectively
E q"'Qmin(a,q)=1) •
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32
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
Since l3(X ,y) = (logX)(I+O(I))A if logy rv A(log2 X)2 / log3 x (A > 0 fixed),
we deduce from Theorem A3 the following result, which sets a limit on the
range of Q for which Bombieri-Vinogradov type estimates for averages of the
form (2.9) may hold.
Corollary 2.2. Let e > 0, A > O. For every x > xo(e, A), there exists an
integer a such that for all Q with
(2.14)
Q
we have
(2.15)
~
(q ,a)=l
le(x; q, a) -
~.
2
x exp( -( 1 - e)A(log2 x) / log3 x),
qJ~)1 ~ I q~2
(e(x; q, a) -
qJ~)) I> x(logx)-A.
(q ,a)=l
The method of proof of Theorem A3 is rather different from that of Theorems
AI, A2. It is based on the following simple idea. We consider primes p rv x,
and for given a we wish to study p == a (mod q) on average over q. This
leads to a double sum over p and q. If we write p - a = qr, then the double
sum can be transformed to a double sum over p and r.
Since our values for q are almost as large as x, the complementary factor r
is quite small, and the transformed sum deals with primes in progressions where
the modulus r is sufficiently small so that analytic methods can be successfully
applied. The resulting estimates lead to a sum that, in the special case that a is
of the form TIp<z P , can be shown to have an oscillatory behavior, which leads
to the conclusion in a manner similar to those of the above methods.
In contrast to the earlier methods, this one offers the advantages of leading
to a stronger result (at least when x/q is not too small), and of not requiring
the rather deep Linnik-Gallagher theorem for its proof. This method seems to
be incapable of dealing with individual arithmetic progressions, however, and
thus cannot even give, as far as one can see, a new proof of Maier's original
result [Ma].
In the case where Q > x(logx)-N for some fixed N, Theorem A3 and the
corollary recover Theorem 1 of [FG]. As in [FG] we deduce lower bounds for
sums I:q Ierror I, by bounding from below the smaller quantity II: q error I .
The integers a produced by the proof grow as x does. We believe that this is
inevitable, at least for bounding II:q error I. That this is true, at least for Q
in the range (2.14), is seen from the following result that slightly refines earlier
theorems of Bombieri-Friedlander-Iwaniec [BFI] and Fouvry [Fo]. The proofs
of these results are based on the same simple switching principle described in
connection with the proof of Theorem A3.
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OSCILLATION THEOREMS
Proposition 2.1. Fix A>
fa
(2.16)
I
o.
33
Thenfor 2 ~ Q ~ !x, 0
(O(X; q, a)- qJ7q))
< lal < (logx)A,
we have
I«A r(a)Qlog(x/Q)+x(logx)-A+Q10glal.
(q,a)=1
Here the last term on the right may be deleted unless a is prime.
Thus, when Q satisfies both (2.11) and (2.14), (2.12) and (2.13) cannot hold
with numbers a± in the range 0 < la±1 < (logX)A .
We conclude this section with some speculation as to what might be the right
upper bound for .1.(x; q, a) and formulate the following:
o.
Hypothesis. Let 8 >
For all q, a, and x with q ~ %(8), (a, q) = 1,
x ~ q(logq)l+e , we have
(2.17)
1.1.(x; q, a)1 «y-(I-e)6(x,y),
where y = x/q and
(2.18)
~(
u
(!
(! ( )) .
. 2' log(logy/log2
X)) -_ mm. 2' U3~ x, Y
x, y ) = mm
1
og2 x
It is equally reasonable to expect that the upper bound given by (2.17) and
(2.18) holds for the short interval analogue where .1.(x; q, a) is replaced by
.1. (x , y) = y-l(O(X + y) - O(x) - y). These hypotheses carry the implication
that the results we obtain on average over q (Theorem A3) and, if GRH be
assumed, the results we obtain here for individual progressions and those given
for short intervals in [HM], are essentially best possible.
As already indicated, the proof of Theorem Aj (1 ~ j ~ 3) is dependent
upon a Theorem Bj, which demonstrates an oscillatory behavior for a certain
sifting function of the type ~n$y,(n,p)=1 f(n). Those theorems are stated and
proven in Chapter II. The proofs of Theorems Bl and B3 depend on a further Theorem C whose proof occupies the greater part of this paper. The final
Chapter III gives the derivation of the above-stated results.
II.
OSCILLATION THEOREMS FOR SIFTING FUNCTIONS
3.
THE STATEMENTS
In this section we state our principal results on oscillations of (J>(y, z) and
certain related functions. These results are proven in §§4-10.
We first introduce some notation. For real numbers y > 0 and z ~ 2 and
positive integers k, we set
(J>k(y, z) = #{n ~ y: pin, p
=y II
p<z
pfk
(1 - .!.)
p
(1
tk
~ p ~ z}
+ rk(y,
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z)),
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
34
and let <I>(y, z) = <1>1 (y, Z), r(y, Z)
form r(P\y, z) of r by
)
r(p (y, z)
= r 1(y,
=j y
dt
r(P)(t, z)-,
y/2
t
where r(P)(y, z) is defined by
n:9
p'ln=}p' ~z, pi oFP
z). We further define a weighted
IT (I-pI1 ) (l+r (P) (y,z)),
n
¢(n)=YC
I
<z
or p'=p
p
with
C
= pQz ( 1 + p' (/ -
1)) ,
p'#
and we set r(y, z) = r(2)(y, z).
Throughout this and the following sections we work with pairs of real numbers (y, z) lying in a range of the type
(Ra)
u
y = z,
z ;?: zo'
Uo :::;
u :::; z
1-a
where 0: is a given positive real number, and Zo and Uo are suitable sufficiently
large constants, possibly depending on 0:. Given y and z, we define u and
!? by
u = logy/logz,
(3.1)
For u > 0 we define
~u
(3.2)
(3.3)
~u
= logz.
as the positive solution of the equation:
/'u =
It is easy to see that
!?
u~u + 1.
satisfies
~u=IOgU+IOg2U+OC:::)
:u~u = ~ + 0 (UI~gu)
(u;?:3),
(u;?: 3)
(cf. [HT, Lemma 1]).
We can now state the two oscillation theorems for the functions rk and r(P)
that are required for the proof of Theorems Al and A3.
Theorem Bl. Let e > 0 and (y, z) be in the range (R 1/ 6+e) with
and let u = log y / log z. There exist numbers z ± satisfying
( 3.4)
min(lz Z1-2/(u+2))
2
'
<
z <z
± ,
such that for every positive integer k satisfying
(3.5)
L
p$.z
plk
1 :::; 210g2 Z,
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Uo
= 1 + e,
OSCILLATION THEOREMS
35
and for some y ± (that may depend on k) satisfying
(3.6)
y(1 - Ijlogy):::; y±:::; y,
there hold
{ rk(y +, z+) ~ exp{ -u~u - c1 u},
rk(y _, z_) :::; - exp{ -u~u - c 1u},
where c1 is a positive constant depending at most on 8.
(3.7)
Theorem B3. Let
8>
0 and (y, z) be in (R 1/ 6H ) with
Uo =
1 +8. There exist
z± satisfYing (3.4) such that for every prime p and for some y± (which may
depend on p) satisfYing (3.6) the inequalities (3.7) hold with rk replaced by
f(P) .
In what follows we are primarily concerned with the condition u > Uo for a
large unspecified uo . How this assumption is relaxed to u ~ 1 + 8 is indicated
at the end of this section. Actually, this weaker condition is necessary for
the proofs of Theorems Al and A3 in that logl+e in (2.3) and (2.12) would
otherwise have to be replaced by 10gUOH .
The above results are quite sharp. It can be shown that, in the range (Re) ,
rk(y, z) is bounded from above by O(exp( -u~u + O(u))) (cf. [HM, Lemma 2]
for the case k = 1). Thus, the lower bound (3.7) is best possible aside from
the value of the constant c1 • Moreover, the localization of z± cannot be much
improved. Indeed, it is clear from Theorem C and Proposition 3.2 that the
numbers u± = logy j log z± cannot be better localized than within an interval
of length 1. On the other hand, the localization of y ± is probably not best
possible. In fact, under the Riemann Hypothesis, (3.7) can be shown to hold
in the range (R 1/ 2+e) for any 8 > 0 with y ± = y (cf. the remarks following
Corollary 3.1). For our applications, however, this deficiency is inconsequential.
We remark that, instead of keeping y (essentially) fixed and varying z in
order to obtain extreme values of rk(y, z) or f(P)(y, z), we could have instead
fixed z and varied y. Specifically, under the conditions of Theorem B 1, it
can be shown that the estimate (3.7) holds with z± = z and suitable numbers
y ± satisfying y :::; y ± :::; Y Z2. A similar estimate, but with a much weaker
localization of y ± ' was obtained in [HM] for the case of r(y, z). This was
achieved by bounding from below certain weighted averages over the function
f(u) = If(zu, z)12. This method, while being relatively simple, is not capable
of yielding results of the quality of Theorems B 1 or B3, which will be needed for
the proofs of Theorems A 1 and A3. Therefore it is necessary to use a different
and more complicated approach.
The restriction (3.5) in Theorem Bl is responsible for the corresponding
restriction (2.1) in Theorem A 1. In order to deal with moduli k containing
many small prime factors, it is natural to consider the analogue to <I>(y, z)
obtained by counting all those n :::; y whose prime factors lie in a given interval.
In general, this should be more difficult to estimate but, in the case that the
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36
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
interval is rather short, a fairly precise oscillation theorem may be obtained by
combinatorial means. The following such result is applied to Theorem A2.
Theorem B2. For z 2:: zo' h ::; !z, k 2:: 1, P the product of any k primes, all
of which are in the interval (z - h , z], we have
(3.8)
(_l)i-lrp(y)~(_l)i-l (
L
1-
n9
rp~)y)
iy(~)z-i
2::
(n,P)=1
for every integer j with 1 ::; j ::; k I 5 and every real y with (z - h)J 2:: y 2::
4jzi l(k-j+l).
We deduce Theorems B1 and B3 from a more fundamental result, Theorem
C, and two auxiliary propositions. Theorem C gives an asymptotic formula for
a "smoothed" version of r(y, z) in terms of the generating Dirichlet series:
F(s)
let
= F(z; s) = IT
p<z
(1 - ~) ((s).
p
To state this result, we require some notation. With F(s) defined above, we
rp(s)
= rp(z; s) = 10gF(z; s),
where the branch of the logarithm is chosen such that rp(a) is real for 0 < a <
1. Such a branch exists and is an analytic function in the region 0 < a < 1 ,
It I ::; 1, since ((s), and hence F(s) , is analytic and nonzero in this region.
Next, given e > 0 and real numbers y, z in the range (Re) , we define So =
so(y, z) as the unique solution to the equation
(3.9)
satisfying
el2 < ao < 1 - 3/g,
(3.10)
and we set
0< to < 2nlg,
p(y, z) = e YF(z; SO)/o-l IsoJnul2,
where u and g are defined by (3.1). (The existence and uniqueness of So
under the stated conditions is shown in Proposition 3.1 below.) Finally, we
introduce a "smoothed" form of r(y, z) by setting
r(y, z; A) =
JX
J7C
/00 r(ye, z)e
v
-Av 2 /2
v 2n -00
where A is a positive real parameter. We then have:
d v,
Theorem C. Let e > 0 be fixed. For (y, z) in the range (Re) and log u ::; A::;
exp(g3/2-e), we have
(3.11)
r(y, z; A) = Re p(y, z) + 0
CPi~~:)I)
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,
OSCILLATION THEOREMS
37
where u and.? are defined by (3.1) and the implied constant depends at most
on e.
For the proof of this result we use the saddle point method. Its basic idea
is to write r(y, z) as a Perron integral over F(S)yS-1 Is along a vertical line
segment whose abscissa is Re so' The main contribution to the integral then
comes from small neighborhoods of the points So and So that are saddle points
of the function F(S)/-I, and integration over these neighborhoods produces
the main term Rep(y, z) in (3.11).
This approach is very similar to the one used by Hildebrand and Tenenbaum
[HT] in the "dual" problem of estimating the function:
I{I(Y, z)
= #{n:5 y: pin => P:5 z}.
In fact, the main term in (3.7) is formally almost identical to the main term in
their asymptotic formula for I{I(Y, z) , except that different Dirichlet series are
involved. There are two important aspects in which the estimation of r(y, z)
differs from that of I{I(Y, z) however. The first is that, while in the case of
I{I(Y, z) there is a single dominating saddle point on the real line, the function
F(S)yS-1 does not have a real saddle point, but rather two saddle points So and
So off the real line whose contributions to the integral are complex conjugates.
(The function has other saddle points as well, but their contribution turns out to
be negligible.) This explains the appearance of the real part function in (3.11).
The second difference between the estimations of r(y, z) and I{I(Y, z) is
that in the case of I{I(Y, z), the generating Dirichlet series can be bounded
trivially on any vertical line by its value at the abscissa. This is not possible
in the case of ·r(y, z) and, as a result, the estimation of the integrand in the
Perron integral at large heights becomes a nontrivial problem that prevents us
from obtaining a formula of the type (3.11) for r(y, z) if u is large, say a fixed
power of z. In order to obtain a better convergence in the Perron integral, we
have to work with the smoother functions r(y, z; A) rather than with r(y, z)
itself.
To complement Theorem C, we prove the following results, which give estimates for the implicitly defined quantities So and p(y, z) in the main term of
(3.11). Here e is an arbitrary given positive number, and the implied constants
depend at most on e.
Proposition 3.1. If (y, z) is in the range (R B ) , then the equation (3.9) has a
unique solution So = so(y, z) satisfying (3.10), and we have
(3.12)
so(y, z) = 1 -
~ + i; (1 + ;u) + O(.?I~g2 u)'
(3.13)
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J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
38
Proposition 3.2. Suppose (y, z) is in the range (Re). Then we have
(3.14)
p(y, z)
= exp{ -u~u + O(u)} ,
(3.15) p(ZU-V, z) = p(zu, z)exp
{V(~u -
in)
+ 0Co~u) + 0(Vu2 )}
(0 $ v $ (1 - e)u),
(3.16) p(zu, z)
= p(zu, z) exp { o(~ (IE(Z,
+ Z;2z + ZB) )
z)1
}
(z $ Z $ 2z),
(3.17)
p(ZU, z)
= p(zu , z) exp { 0(uI2) }
where
L
E(z, z) =
(Z
~ z),
logp - (z - z).
Theorem B2 is proven in §4. Propositions 3.1 and 3.2 and Theorem Care
proven in §§5-8. In §§9 and 10 we deduce Theorems Bl and B3 from Theorem
C and the two propositions. We conclude this section by deriving two simple
corollaries of Theorem C and then reducing Theorems Bland B3 to the range
u;::: uo ' where Uo is arbitrary. First we show that the estimates of Theorem C
hold with r(y, z) in place of r(y, z; A) , provided u = log y I log z is not too
large.
Corollary 3.1. Let e > 0 and (y, z) be in the range (Re), and suppose that
u$ 2
3/ 2 - e .
Then we have
r(y, z) = Re p(y, z)
+ O(lp(y,
z)11 log u).
To deduce this result, we apply Theorem C with e replaced by el2 and
A = exp(2(3-e)/2). Without loss of generality, we may assume that e < 1/2.
Using the estimates:
r(ye V , z)
and
= r(y,
z)
+ O((lvl + lly)2)
r(y' , z) = 0(2)
(Ivl $ 1)
(y' > 0),
which easily follow from the definition of r(y, z), we get
r(y, z; A) =
r(y~v'X [II e-).v /2 dv + 0(2v'X [~I
2
+o
(2 VI foo
(IVI
+ ~ )e-).v 2 /2 dv )
e-).v 2 /2 dV)'
The error terms here can be bounded by
« 2 + 2 +2e-)'/2« exp{_u1H/3}« p(y, z)
v'X y
logu '
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39
OSCILLATION THEOREMS
if we use the definition of A, the bound u ::=; ,2>3/2-e , and the estimates (3.14)
and (3.2) for p(y, z) and C;u' Moreover, the integral in the main term equals
ill e-).v 2 /2 dv
=
V;;
(1 + O(e -)./2))
Thus,
r(y, z; A) = r(y , z) ( 1 + 0
= ~n
Co~ u) )
(1 + O(~
+0
) ).
CPi~~:) I)
and the estimate of the corollary follows from (3.11).
We note that under the Riemann hypothesis, A can be taken as large as
exp(zl/2-e) in Theorem C, and the estimate of the corollary can be shown to
hold in the range (R I / 2+e) for any e > O. By appealing to Corollary 3.1 instead
of Theorem C in the proof of Theorems B1 and B3, one can then show that,
under the Riemann Hypothesis, one can take y± = Y in (3.7), as we remarked
earlier.
Our second corollary concerns the asymptotic behavior of the Buchstab function w(u), defined as the continuous solution of the system
w(u)=lju
(1::=;u::=;2),
(uw(u))' = w(u - 1)
(u> 2).
This function was introduced by Buchstab [Bu] who showed that
lim r(zu, z) = eY(w(u) - e- Y)
(3.18)
z-+oo
holds for every u > 1 , where y is Euler's constant. Buchstab [Bu] also obtained
the estimate:
w(u) = e- Y+ O(exp{-uc;u + O(u)}).
In view of this relation it is natural to consider the function
W(u) = w(u) - e- Y.
Maier [Ma] showed that W(u) changes sign in every interval of length 1. Cheer
and Goldston [CG] proved that W(u) has at most two sign changes in any
interval of length 1. Furthermore they showed that
max W(v) =
v~u
max W(v),
u~v~u+2
min W(v) =
v~u
min W(v).
u~v~u+2
Theorem C and Proposition 3.2 enable us to describe the behavior of W(u)
more precisely as follows.
Corollary 3.2. There exists a decreasing function Wo(u) satisfying
(3.19)
Wo(u) = exp{-uc;u
+ O(u)}
(u 2: uo)
and a function 0 (u) satisfying
O(u + t) - O(u) = tn(l
+ O(ljlogu))
(u 2: Uo ' 0 ::=; t ::=; 1)
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40
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
such that, as u
-+ 00 ,
W(u) = Wo(u){cosO(u)
+ O(ljlogu)}.
This result clearly exhibits the oscillatory behavior of W(u). Aside from a
"decay factor" UiQ(u) , the function W(u) behaves like a trigonometric function
with period 2. An immediate consequence of this behavior and the estimate
(3.19) is that
max W(v) = exp{ -u~u + O(u)}
u~v~u+2
and similarly,
min
u~v~u+2
C,
W(v) = - exp{ -u~u
+ O(u)}.
To prove the corollary, we note that, by Buchstab's result (3.18) and Theorem
W(u) = e- Y lim r(zu, z)
z--+oo
= e- Y lim {Rep(zu, z) + O(lp(zu, z)ljlogu)} ,
z--+oo
provided u is sufficiently large. Also, the estimates (3.17), (3.14), and (3.15)
of Proposition 3.2 imply that the limit
l(u) = lim p(zu, z)
z--+oo
exists for every sufficiently large u and satisfies
II(u)1 = exp{ -u~u
+ O(u)}
and
argl(u
+ t) -
argl(u) = tn(1
+ O(ljlogu)),
(u ~ uo' 0 ~ t ~ 1),
where argl(u) denotes a continuous branch of the argument of l(u). Taking
Wo(u) = e-YII(u)1 and O(u) = argl(u) then yields the asserted estimates.
As already mentioned, the results of Theorems Bland B3 follow fairly easily
in the range 1 + e ~ u ~ Uo for fixed e and uo ' In this range the formula
(3.18) of Buchstab still holds with r replaced by rk (subject to (3.5)). In the
case u ~ 2, this is an immediate consequence of [FG, Lemma 5]; in the case
1 + e ~ u ~ 2, that proof needs only a slight modification. An easy variation
of Buchstab's original arguments shows that (3.18) also holds with r replaced
by r(P) • Theorems Bl and B3 then follow for 1 + e ~ u ~ Uo from these facts
in conjunction with Maier's result that W(u) changes sign in every interval of
length 1 [Ma, Lemma 4].
Note. We take this opportunity to correct an error in [FG]. Professor Warlimont
has kindly pointed out to us an omission from the identity in the proof of
Lemma 5 in [FG]: specifically that the contribution "'£~/y<n~x 1 was overlooked.
As before, ",£' denotes a sum over integers all of whose prime factors divide q,
and q is assumed to have fewer than / distinct prime factors where f = f(y)
tends to zero as y -+ 00 with f(y) > 310g2 y j log y .
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OSCILLATION THEOREMS
41
In order to complete the proof of Lemma 5 it suffices to establish, for x ;:::
y ;::: Yo' the estimate
~' 1 «
(3.20)
<
n_x
q
3
xf
+ -(-)
-'
rp q 1
ogx
xl log y
By first considering the contribution to the sum of those n :::; XI/2 , then those
n divisible by some prime >
and finally the remaining n, we find that
(compare Exercise 01 of[HaT]) for (1 = II flogy,
if ,
",'
L.J 1 «x
1/2
,"
1/2 u
+ v(q)xly 2f + "L.J
(nix )
n~x
(3.21)
xlf2<n~x
« x/log3 y + X -u/2 ~* n
U ,
n~x
where E* denotes the restriction to those n divisible only by prime divisors
of q that are :::;
By a Theorem of Halberstam and Richert [HaT, Theorem
00], we have
if .
",*
(3.22)
~
nU
« ~ ",* _1_ «
logx ~ n l -
«
u
_q_ ~ exp { ' "
rp(q) logx
pL.J~y2J
(_1__ P.!.) }
pl-u
q
x
rp(q) logx'
Therefore (3.21) and (3.22) yield (3.20), since x- u / 2
Lemma 5 follows as stated.
«
flogy/logx, and
4. PROOF OF THEOREM B2
For m a positive integer, let
s(m)=~.!.,
v(m)=~1.
pimP
plm
We require the following lemma:
Lemma 4.1. For any positive integers m, j with m squarefree and s( m) :::; j 12,
we have
~
(4.1)
dim
.!. < 2s(~)
d -
]
v(d)~j+1
~
dim
v(d)=j
.!..
d
Proof. We first show that, for every integer i ;::: 0,
(4.2)
~
dim
v(d)=j+i
1
j!
i
1
(j:::; ( . .),s (m) ~ (j'
]
+l
•
dim
v(d)=j
We observe that any divisor d of m with v(d) = j
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+ i can be written in
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
42
the form d = fSPI .. 'Pi (where v(fS) = j and the Pf.l are prime) in eji)i! =
(j + i)!1 j! distinct ways, if we distinguish amongst different orderings of the
Pf.l'
By multiplying out and collecting terms on the right side, we get (4.2). To
prove (4.1) we note that the left side is obtained by summing over i the left
side of (4.2). The application of (4.2) thus yields
"
L...J
dim
lI(d)~i+1
1
" 1 , , j!
i
d:5 L...J d L...J ( . + i)!s (m)
dim
i~1 )
v(d)=i
:5
2: ! 2: e(~))i,
dim
lI(d)=i
d i~1
}
and since s(m):5 j/2, this gives (4.1).
Proof of Theorem B2. If diP with v(d) ~ j, then d > (z - h)i ~ y, so
[yld] = O. Now, since
rp(Y) =
2:
1-
n9
(n,P)=1
diP
we have
(4.3)
{O~)y = 2:1l(d)([~] -~),
2: ~1:5Y
Ir p(y)+(-l)i y
diP
lI(d)=i
1
-+
d
2:
diP
lI(d)~J+1
1.
diP
lI(d)~J-1
Since s(P) :5 n(z)/(z - h) :5 1/8 for z > zo' we may use Lemma 4.1 to
estimate the first term on the right-hand side of (4.3). Since C~I) :5 H~) for
i :5 k I 3, the second term is
:5
~(~)
:5 ~ 2: ~.
2z }
diP
lI(d)=J
Substituting these in (4.3) we complete the proof of Theorem B2.
5. PROOF OF PROPOSITION 3.1
We begin with a lemma, which is needed again in later sections.
Lemma 5.1. Let e > 0 and k ~ 1. Uniformly for z ~ 2 and s
range
(5.1 )
e :5
(J
< 1 - 1I:? ,
It I :5
1,
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= + it
(J
in the
43
OSCILLATION THEOREMS
we have
(5.2)
r(1-S).'l'
tp(z;s)=- 11
(5.3)
(5.4)
(5.5)
tp'(z; s)
tp" (z; s) =
Itp
III
=
eW
w dw + O
(
1-0')
~k
+0(1),
(1 + O(;k)) ;~: + 0(1),
(1 + o(~)) zl-s(1 -
(1 ~ s)~)
(1 - s)
~
(z; s) 1:$
(
+ 0(1),
1-0'
1
2
)) Z
1 + 0 . (1 _ (J)~
1 _ (J ~ ,
(
where ~ = log z and the implied constants depend at most on e and k.
Proof. Differentiating tp(z; s) = 10g(TIp<z(1 - Ijp s),(s)) , we obtain
'( ) _ ' " logp
tp s - L.,.. ~
p<z p
(5.6)
+
" (s)
r(s)'
..
In the range (5.1) we have
,'(s)
'(s)
and
' " logp
L.,.. S
1
p<z p -
= _1_ + 0(1)
1- s
= '"
logp 0 ('" 10gp)
L.,..
+ L.,.. 20'
p<z p
p<z p
S
1 ))
= ( 1 + 0 ( ~k
1 ))
= ( 1 + 0 ( ~k
du
1rI 17
+ O( 1) + O( z 1-20' log z)
Z
I-s
1_~
1
+ O( 1) ,
using partial summation and the prime number theorem with logarithmic error
term. Substituting these estimates into (5.6) yields (5.3). The estimates (5.4)
and (5.5) can be proved in the same way.
To obtain (5.2), we note that by (5.3),
(
tp(z; s) - tp z; 1 -
1) l
~
=
s
1-1/.'l'
tp , (z; s , ) ds ,
1 r(l-s).'l' , (
w)
=-~11
tp z;l- ~ dw
=-
l
(I-S).'l'
1
eW
-dw
w
+ O(~~+ll(I-S).'l' I~ IldW1) + 0(1)
=-
r(I-S).'l'
11
eW
w dw + O
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(
1-0')
~k
+0(1).
44
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
Since
9'(Z; 1-
~) = ];log (1- pl_11/.'? )
= -
+IOg,(l-
~)
1
L
I-I/.'? + 10g2' + 0(1)
p<zP
=- L
-1 + log2
p<zP
Z
+ O( 1)
= O( 1) ,
this implies the desired estimate (5.2).
We now proceed to prove Proposition 3.1. We assume that (y, z) lies in the
range (Re) with a given parameter e > 0 and sufficiently large constants Zo and
uo ' We first establish the existence and uniqueness of so(y, z) as a solution
to (3.9) in the range (3.10). To this end it suffices to show that (i) for every
a E [e/2, 1- 3/2'], there exists a unique number t = t(a) E (0, 2TC/2') such
that 9'/(a + it) is real and negative, and (ii) the function f(a) = 9'/(a + it(a))
is strictly increasing in the interval [e/2, 1 - 3/2'] with f(e/2) < _zl-e2'
and f(1 - 3/2') = 0(2').
For the proof of (i) we observe first that, by (5.6), 9'(a) is positive for
o < a < 1, since ((a)/'(a) > 0 in this interval. The assertion (i) follows if
we can show that
(5.7)
holds for e/2::; a ::; 1- 3/2' and 0::; t ::; 2TC/2'. Using Lemma 5.1 with e/2
in place of e and k = 1 , we see that in this range
d
/
.
d
/.
9'''(a+it)
-d arg9' (a + It) = -d 1m log 9' (a + It) = Re /(
.)
t
t
9'a+lt
= Re ( 1 + 0
(~ ) )
C~
s - 2' )
_(1 + 0 (_1
)) ((1_a)2+t
1- a 2 _ 2') .
2'
Since
0<
I-a
1
2'
<--<(1 - a) 2 + t 2 - 1 - a - 3
for 0 < a < 1- 3/2' , (5.7) follows if z (and hence 2') is sufficiently large, as
we may assume. Since arg 9'/ (a + it) is a continuously differentiable function
of (a, t) in the range under consideration, the function t(a) defined by (i) is
continuously differentiable as well.
To prove (ii), we note that the function
I
(a) = 9''' (a
+ it(a))(1 + it' (a))
is real-valued, continuous, and has no zeros in the interval [e/2, 1-3/2'], since
by (5.4), 9''' (a + it(a)) =I 0 in this interval. Thus f(a) is either monotonically
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OSCILLATION THEOREMS
45
decreasing or monotonically increasing, as a increases from e/2 to 1 - 3/£7 .
By Lemma 5.1 and the definition of t(a), we have
=
+ it(e/2))
-Irp' (e/2 + it(e/2))1
=
11-e/~-it(e/2)1
f(e/2) = rp' (e/2
l-e/2
(
(
1 ))
1+0 £7
'
which is < _zl-e£7 , if z is sufficiently large. Similarly,
f( 1 - ;
)
= 1(3/£7) ~~::;_ 3/£7)1 (1 + O(~
which is of order 0(£7). Assertion (ii) now follows.
It remains to prove estimates (3.12) and (3.13). Setting
Sl = a l + it l , we can restate (3.12) as
al
(5.8)
= eu + 0(_12_ ) '
log u
From (3.9) and (5.3) we get
-u£7
= rp' (z;
so)
tl
)),
So =
1 - sl/£7 ,
= -n(1 + } ) + 0(_12- ) ,
"u
= (1 + 0(~2
log u
)):~I £7 + 0(1),
so that
( 5.9)
Since
lesil = /'1
and
ISII =
lal + itll
=
al (1 + O(t~/a~)) = al (1 + O(I/a~)),
we deduce from (5.9) that
e (11/ a l xu,
which implies a l »logu and, by another application of (5.9),
(5.10)
~I
=
=
u( 1 + 0(~2) + O(u1 ) + O(:~))
u(1 + 0 CO;2 u) ).
Now note that the function eX / x is monotonically increasing in x 2:: 1 and
that, by the definition of u '
e
/.+0
(5.11)
eu +J
(ue u + 1)(1 + c5 + 0(c5 2 ))
=
eu (1+ c5 le u )
=u(I + (1 -
I/e u )c5
+ 0(c5 2 ) + O(I/ueu ))
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(1c51::; 1).
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
46
Comparison of (5.10) and (5.11) shows that a, = ~u + ~ with ~ = O( 1j1og2 u) ,
as asserted in (5.8).
To prove the second estimate of (5.8), we take the argument on both sides of
(5.9), keeping in mind that by (3.10), t, = -to.:? lies in the interval (-2n, 0).
This yields the equation
1) + 0 (-1 ) = t - arctan -t, = t - -t, + 0 ( -1 ) ,
-n + 0 ( -CL72
u.:?'
0"
0'
".2
..z;
"
v,
which implies the desired estimate
t, =
-n(l +~)
+ 0(_1
2 ) -n(l + ; ) + 0 ( - \ ).
a,
log u
"'u
log u
=
Finally, to obtain the estimate (3.13) for the derivative of so' we differentiate
both sides of the equation (p' (z; so) = -u.:? with respect to u. In this way,
we get
so that
()
u
£lUSO(Z ,z) = u
.:?
"( .
rp z,
So
)
(1 +o(.:?1))
=rp' (z1; so) (1+0( (1 - o'o).:?
1
))= __
1 (1+0(_1 ))
u.:?
log u
_
(1-s o)2.:?
- z'-SO((l - so).:? - 1)
by the estimates of Lemma 5.1 and (3.12). This proves (3.13).
6.
PROOF OF PROPOSITION
3.2
First we consider the behavior of p(y, z) when z is fixed and y =
Setting
u
j(u) = logp(z ,z) = log
(6.1 )
=
ZU
varies.
eYySo-'P(s)
So
..fiiU720
nu/2
(so - l)u.:? + rp(so) -logso -
eY
1
2 10gu + log In/2'
we can rewrite the first two estimates of Proposition 3.2 as
j(u) =
(6.2)
-u~u
+ O(u)
and
(6.3)
j(u - v) = j(u) +
v(~u -
in)
+ 0Co~u) + 0(Vu
2
)
(0 ::; v ::; u(l - e))
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47
OSCILLATION THEOREMS
respectively, and it suffices to show that these latter two estimates hold for
Uo ~ U ~
z 1-8
Taking the derivative with respect to u in (6.1) we obtain
'07 + (so - 1)..z;0+7
' " - so/so - Ij2u,
/ (u) = sou..z;
rp (so)so
where for brevity we have set s~ = (%u)so(ZU, z). Since rp'(so) = -u£' by
(3.9) and
(so - I)£' = -C;u + in + O( 1/log u) ,
;~ = u~so (1 + O(IO~U)) = OCo~u)
by Proposition 3.1, it follows that in the range (Rs)
/(u)
(6.4)
This implies, for 0
~
= -C;u + in + 0(1/10gu).
v ~ (1 - e)u,
f(u-v)=f(u)= f(u) +
fu~v/(W)dW
fu~v (C;w -
in + O(IO~W) ) dw
= f(u) + v(C;u - in) + 0(v 2 /u) + O(v / log u),
and hence (6.3), after adjusting the constant Uo in (Rs) , if necessary. Here we
have used the estimate
C;w = C;u + O(lu - wl/u)
(Iu -
wi
~
(1 - a)u),
which follows from the bound (3.3) for (djdu)C;u.
To prove (6.2), we apply (6.4) again to obtain
f(u)
= f(u o) +
f" / (w) dw
1uo
= f(u o) - f"(C;w + O(I))dw
1uo
= f(u o) - uC;u + O(u) ;
then (6.2) follows, if we can show that f(u o) = 0(1). Since, by Proposition
3.1,
(so - I)£' = -c;u + 0(1),
we see from (6.1) that f(u o) = rp(so) + 0(1), where So
Lemma 5.1 and Proposition 3.1, we have
{(I-So)..'? eW
11
(ZI-crO )
= so(ZU
W dw + 0 £,2 + 0(1)
= O(e(l-cro)..'?) = O(iUo) = 0(1).
rp(so) = -
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O,
z). Now, by
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
48
Hence we obtain f( uo) = O( 1) as desired.
It remains to prove estimates (3.16) and (3.17). To this end we consider
p(ZU, z) as a function of z with u fixed, and define, in analogy to (6.1),
(6.5)
g(z) = log p(zu , z) = (so - l)u log z + <p(z; so) -logso + log
el'
J1iU72'
nUl2
Setting So = so(ZU , z) and So = so(,zu , z) , the asserted estimates can be restated
as
(6.6)
Ig(z) - g(z)1
«
iIE(z, z)1
+
;2
+ z~-e
(z ~
z ~ 2z),
and
(6.7)
where
Ig(z) - g(z)1
~
= (z -
« ul£1
(z
~
z),
z)lz. We shall prove that (6.6) holds in the case
(6.8)
The relation (6.6) for £1- 3 ~ ~ ~ 1 and the relation (6.7) can thus be
deduced by splitting the interval (z, z] into subintervals of the form (zl'
zl (1 +£1- 5 )] (plus possibly one interval of shorter length) and by applying (6.6)
(with e replaced by e12) along with the prime number theorem in the form
E(ZI(l +£1- 5 ), zl)« zl£1- 7 to these intervals.
From (6.5) we obtain
where
= u(so - so) log z + <p(z; so) R2 = <p(z; so) - <p(z; so) + u( 1 R3 = 10g(501so),
Rl
<p(z; so) ,
so) log(z I z) ,
To prove (6.6), we show that each of the terms R j is bounded by the right-hand
side of (6.6).
We first derive an estimate for So - so' By the definition of So and so' we
have
with
~)
R 4 = <p '(~~)
z; So - <p '( z; So
R5
=
'"
~
logp
-s--,
z9<2 P
0 -
1
~,
ISO <p"(z; s)ds.
= <p ,(z; so)
- <p (z; so) =
So
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49
OSCILLATION THEOREMS
Since
«
ZI-aO(A + Ze-I)
« e~u (A + ze-I) « u(log U)(A + ze-I)
by a standard upper bound for primes in short intervals and Proposition 3.1, it
follows that
IRsl « IR41
(6.9)
+ ulog(z/z)
« u(log U)(A + z
e-I
).
On the other hand, since by Proposition 3.1 and our assumption (6.8),
(6.10)
so that
IRsl =
IlsO fIJI! (z; s) dsl » u.2"2 lso - sol.
Comparing this bound with (6.9), we deduce that
(6.11)
~
log u (
Iso-sol« .2"2 A+
1)
zl-e
•
Since R3 = log(so/so) = O(lso - sol), R3 is bounded by the right-hand side of
(6.6).
To estimate R 1 , we write
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50
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
and we note that by (3.9), the first term on the right-hand side is equal to
u2'(.s0 - SO). It follows that
by (6.11), (6.10), and (6.8). This proves the desired bound for RI .
It remains to estimate R 2 • We have
R2=U(I-.s0)10g~(6.12)
= u( 1 - .so) log ~ +
+
z
o( L
z$.P<z
L
z'5,p<i
10 (lg
p
~o)
. 11
"logp
L..J
ogz z'5,p<i
NSo
Z
N
1zNLOgp
N - ~o I) + O(lz - z + llz- 2uO ).
log z p
The second error term here is of order
«
z2(I- uol
(~ + ~)
Z
«(ulogu) 2
z2
(Az + 1) «u27
2
Z2
(
A+
z1)
by Proposition 3.1 and the hypothesis u ~ zl-e , and the first can be bounded,
using (6.8), by
"
1og(N/)
« zN-UO L..J
z z «zNI- uo z-z+l 1og(N/)
z Z
N
z'5,p<i
Z
AI) A Au log u AU log u Au
« (1
u og u) ( L1 +
L1« 23 +
z
« 2 2'
z
Moreover, since by Lemma 5.1and (3.9),
NI-s
zl-so = IZ
:
- So
(I - .so)
= (I + 0(1/2 2)),P'(z; .s0)(1 = -(I
.so)
+ 0(11
+ 0(1/22) + 0(I/u2))u(1
- .sol)
- .so)(logz) ,
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OSCILLATION THEOREMS
51
we have
= -u(1 - so) (lOg ~ + E(Z;
+0(,1
Z)) + 0 ( ; )
+ 0(Uil2)
~olil) +O(z~-e).
Inserting these estimates into (6.12), we obtain
u
~
IR21 « zIE(z, z)1
uil
2
+ 22 + uil +
11 - So lil
2
u
+ zl-e '
which proves the desired bound (6.6) for R2 in view of (6.8), and the fact that
11 - sol «eu / 2 « 10gu/2 by Proposition 3.1 and (3.2).
Thus the proof of Proposition 3.2 is complete.
7.
PROOF OF THEOREM
C:
PRELIMINARIES
In this section we derive estimates for the functions F(z; s) and rp(z; s)
that are needed in the proof of Theorem C. We first state two auxiliary results
from prime number theory.
Lemma 7.1. Let 0 < e :::; 1 be fixed.
0:::; a < 1, and It I :::; exp((10gz)3/2-e),
Then we have, uniformly for z 2:: 2,
1
(I-a
)
2: A()
n~ =zl_~ +0 :_a exp(-(10gz)e/2) +0(1).
I-s
n<z
Proof. This result is stated, in a slightly weaker form, in [HT, Lemma 6]. It can
be established in a standard manner by writing the left-hand side as a Perron
integral over the function " /, and then shifting the line of integration into the
critical strip, using Vinogradov's zero-free region. We omit the details.
Lemma 7.2. For any fixed e > 0 and uniformly for a 2:: e, It I 2:: 1, we have
IC(a + it) I «max(l, Itl l - a ) 10g(1
I,'(a
+ it) I «max(l,
+ It I) ,
+ Itl).
Itl l - a )10g2(1
Proof. The first estimate is well known; see, e.g., [Iv, Theorem 1.9]. The second
estimate follows from the first by Cauchy's integral formula.
For the remainder of this section we let z 2:: 2 be fixed and write F(s) =
= rp(z; s), and 2 = log z. The next three lemmas give upper
bounds for these functions for various ranges of values s.
F(z; s), rp(s)
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52
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
Lemma 7.3. Let 0 < e :::; 1 be fixed. Then we have, uniformly for a
It I ~
~
e and
1,
IF(s)l<< {2'lt ll /.2'IOg(1 + Itl)
2'ltl l -O' log(1 + It!) exp(2z l -0' /(1 - a)2')
Proof. Lemma 7.2 yields
IF(s)1 =
I((S) 11 (1 -
:::; 1((s)1
ifa ~ 1-1/2',
ife:::; a < 1 - 1/2'.
;s) I
IT (1 + p~)
p<z
« max(1, It I1-0' ) log(1 + It I) IT (1 +
p<z
In the case a
~
(1f )
p
.
1 - 1/2' , the desired bound follows from the estimate
In the remaining case e :::; a :::; 1 - 1/2' , we obtain the result from the estimates
IT (1 + ~) «
p<z
p
1((a)(1 - a)
IT (1 _ ~) I-I =
p
p<z
exp( ~ ~e:(a))
:::; 2' exp(lqJ(a)l)
and (cf. Lemma 5.1)
1 ))
IqJ(a)1 = ( 1 + 0 ( 2'
1 ))
:::; ( 1+0 ( 2'
2z l -0'
:::; (1 - a)2'
r(I-O').2'
JI
eW
W dw + 0(1)
1-0'
(
1-0'
(l~a)2'+O (1~a)22'2
)
+ 0(1).
Lemma 7.4. Let 0 < e :::; 1 be fixed. Umformly in the range
(7.1)
e :::; a :::; 1 - 1/2',
1 :::;
It I :::; exp(2'3/2-e),
we have
(7.2)
IF(s)1
«
Itll-O' log2(1
+ It I) exp((1 + e)zl-O' /2'),
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+0(1)
OSCILLATION THEOREMS
53
Proof. Setting
",(s) = Z)og (1 - ~) ,
p<z
p
we have, by Lemma 7.2, in the range (7.1)
IF(s)1 = 1,(s)1 exp(Re ",(s)) « Itl 1- oo 10g(1 + It!) exp(I",(s)l).
Thus, to prove (7.2) it suffices to show that
+
+
(7.4)
1",(0- it)1 :5 (1 e)zl-oo /2'
holds in this range. Similarly, since
1F'(s)1
+10g2(1 +It I) +0(1)
= I"(S) II (1 -~) + C(s) II (1 -~) L
p<z
p
«ltI 1- oo loi(1
p<z
p
1?::11
p<z p
+ It I) exp(I",(s)l) (1 + 1?;;?::1 /)
by Lemma 7.2, (7.3) is seen to be a consequence of (7.4) and the bound
~ logp I
1-00
-S-1 «z .
I L.....t
p<zp -
(7.5)
Lemma 7.1 yields for the range (7.1)
I: l?gPI = L A(~) + O(I: 10~;) + 0(1)
n<z n
p<z P -
p<z p
= Z11-~~ 1 + O( ;~: exp( _2'8/2)) + 0(1).
Since 11 - sl ~ It I ~ 1 and 1 - 0- ~ 1/2' by (7.1), this gives (7.5). The same
estimate shows that for 0- :5 0-' :5 1 - 1/2' , we have
1""(0-'
+it)1 = !?; p)~~~ 11 :5 (1 +O(~ ) )Zl-oo' +0(1).
This implies that
k(o-+it)-
",(1- ~+it)!:5 (1 +O(~)) i 1zl-oo' do-' +0(1)
:5 (1 + (~
z;00 + 1
)
))
0
O(
1-00
Z
:5 (1 + e) 2' + 0(1),
and since
I", (1 - ~ +it) I I?; pl-l~2+it 1+ 0(1) I~ pl+l~2+it 1+ 0(1)
= 110g ,( 1 +~ +it) 1+ 0(1) :5 log2(1 +It I) +0(1)
=
=
by Lemma 7.2, (7.4) follows.
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54
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
For the last lemma of this section we let e be a given positive number; y
and z fixed numbers in the range (Re); and u = logy/log z. We further let
So = so(y, z) = 0'0 + ito be defined as in the theorem by (3.9) and (3.10). The
implied constants are allowed to depend at most on e.
Lemma 7.5. For S =
(7.6)
0'0
+ it, It I :::; 1, Iltl- tol > 1/2, we have
IF(s)1
«
2
IF (so) Iexp( -u/310g u).
Moreover, we have
(7.7)
IF(so)1
= exp{u(l + O(l/logu))}.
= 0' + it
Proof. Lemma 5.1 yields, for s
as in the lemma,
F(s) = exp(qJ(s)) = exp { - Ei((l - s)2)
where
+
o( -:;:) +
0(1)},
Ei(w) = fW eV dv
11 v
is a variant of the exponential integral. By Proposition 3.1 and (3.2) the first
error term can be bounded by
ZI-uO
lou u log u
U
--4 «-4« --4- « -3-'
2
2
2
log u
Thus, the estimates (7.6) and (7.7) are seen to be equivalent to
(7.8)
Re{Ei((1 - so)2) - Ei((1 - s)2)}:::; -u/310g2 u + 0(1)
and
(7.9)
Re(Ei(1 - so)2) = -u(1
+ 0(1/logu)),
respectively, and it suffices to prove the last two estimates.
We use the easily derived expansion
(7.10)
.
El(W) = e
w(1-
W
1 2 (1))
4'
IRewl
+ 2' + -3 + 0
which holds uniformly in Rew
gives
W
~
w
1. Applying (7.10) with w
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= (1
- so)2
OSCILLATION THEOREMS
55
Now,
(7.12)
by Lemma 5.1 and (3.9), and since
I( 1 - so)2'1 x (1 - (Jo)2' x
r;u x
log u
by Proposition 3.1, we see that (7.11) implies (7.9). A further application of
Proposition 3.1 shows that
and
Re
1
(1 - so)22'2
=
1
(1 - (JO)2 £72
+0(_1_)=2..+
0(_1_)
10g3 U
10g3 U .
r;~
Therefore, from (7.11) and (7.12) we obtain the estimate:
(7.13)
Re(Ei((1 - so)2'))
= -U(1 + .; +
22
"u r;u
+
0(_13-)).
log u
Next we apply (7.10) with w = (1 - (Jo - it)2' ,where It I :::; 1. Estimating
each term on the right-hand side by its modulus in the same way as before, we
obtain
(7.14)
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56
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
Thus the comparison of (7.13) with (7.14) shows that
Re(Ei((I - so)£'))
+ IEi((1
- s)£')1
~ -~ +
2~u
0(+),
log
u
and hence (7.8), holds in the range to + I/£' ~ It I ~ 1.
Finally, to prove (7.8) for It I ~ to-I/£', we apply (7.10) with w = (1-s)£'
to obtain
Re(Ei((I - s)£')) = Re (1 zl-;£, +
- s
= (1
o( (1 - ZI-<1~£'2)
00)
~)::;£' cos(t£') + 0Co~u)
= u( cos(t£') +
0Co~u))'
Since -cos(t£')~O.9+0(I/logu) for It I ~ t o-l/£'=(1l-I+O(I/logu))/£',
this implies that
- Re(Ei( (1 - s )£'))
~ I~ u ( 1 + 0 (10~ u ) ) ,
which, in conjunction with (7.13), proves (7.8) for the range
8.
PROOF OF THEOREM
C:
It I ~ to -
I/£'.
COMPLETION
We begin by expressing r(y, z) and r(y, z; 2) as Perron integrals over the
functions F(s) = F(z; s).
Lemma 8.1. For any fixed e > 0 and uniformly for y
and T;::: 1 , we have
1
(8.1 )
r(y, z)V(z) = - .
2m
l<1+iT
<1-iT
> 0, z;::: zo' e < 0 < 1 ,
(10 -)
F(s) s-)
Y
ds+ 0 ~
s
T
1)
lOgy) + 0 (( 1 + Y Te/2
£'
(2Z1-<1))
+0 (Y
exp (1 _ o)£'
'
where V(z) = II p <z(l- l/p), y = maxey, 3), £' = logz, and the implied
constants depend at most on e.
Proof. Since the right-hand side of (8.1) is a continuous function of y, and
since
1
r(y, z) V(z) = -<l>(y, z) - V(z),
y
as a function of y, has jumps of order at most O(l/y) , it suffices to prove
(8.1) in the case where y is not an integer. Perron's formula then gives, for
any 0) > 1,
1
f<1 l +iT F(s)yS
(
<l>(y, z) = 21li J<1 I -iT - s - ds + 0
00
(y)<11
~ Ii
1)) .
. (
mm 1, Tllog(y /n)1
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57
OSCILLATION THEOREMS
Taking a1 = 1 + l/logy, the error term becomes
«
y+ 1 ~ 1
-rL- nUl +
n=1
« y +1
T(a 1 -1)
~
L-
y/2<n<2y
. (
mm 1,
y)
TI - nl
y
+ y 1ogy- + 1 « y 1ogy- + logy.
T
T
Replacing the line of integration by the line [a - iT, a + iT] introduces a
residue yV(z) from the pole of F(s) = '(s) ilp <z(l - lips) at s = 1, and an
error term that is of order
«
«
UI-iT lul+iT)/F(S)YS/
(lu-iT
+
- - Idsl
u+iT
S
y;
1 ,;!(IF(a' + iT)1 + IF(a' - iT)!)
y + 1 l-u+l/.2"
( 2z 1- u )
« ---;yT
log( 1 + T)g exp (1 _ a )g
«
y+1
(2Z1-U)
Te/2 g exp (1 - a)'!?
by Lemma 7.3. Thus we have
yr(y, z)V(z) = rp(y, z) - yV(z)
=~
r+iT F(s)yS ds + O(YIOgy)
2m Ju-iT
S
T
+ 0(YT~21 gexp C12~;)~
)),
+ O(logy)
from which (8.1) follows.
For the remainder of this section, we fix a positive number e, which we can
assume to be sufficiently small, and real numbers y and z in the range R e ,
and we let So = a o + ito be defined by (3.9) and (3.10). Thus, in particular, we
may assume that y, z, and u = logy I log z , are larger than any given constant.
We also fix a real number), in the range
2 :'5 ), :'5 exp(g3/2-e) ,
(8.2)
and define the function
k(s)
(8.3)
= k;.(s) = exp(s 212),).
In the sequel, the constants implied in the symbols " 0" and " «
to depend on e, but not on y, z , or A..
Lemma 8.2. With To
(8.4)
= y3/B
" are allowed
we have
1 luo+iTo F(S)/-1
(lOgy)
r(y,z;A.)V(z)=-2'
k(s-l)ds+O - .
7tl uo-iTo
S
Y
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58
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
Proof. Applying (8.1) with (J = (Jo' T = To, and ye V in place of y, we obtain
~
r(y, z; A) V(z) =
=
v 21t
1
10'0+iToF(s) Y s-I k(s -
~.
1tl
where
k( s- 1)
roo r(ye V, z) V(z)e- Av2 /2dv
1-00
O'o-ITo
V1
=.~
v 21t
1
00
-00
S
1) ds +
°(3L Ri )'
i=1
V(S-I)-AV 2/2 d
(S-I)2/2).
e
v=e
is defined as in the lemma by (8.3),
roo log(yev) e _AV /2 dv
RI = VI
1
1-00
.r;: 00
«VA.
-00
2
To
logy + eV -Av 2 /2 d
logy
T.
e
v«3jB'
0
y
roo log(y?) e _AV /2 dv
ye
«VIlogy roo (1 + eivi)e-AV2/2 dv «
R2 = VI
y
and
R3
V1
= T;/22 exp
2
2
1-00
« l/2 exp
1-00
(2ZI-0'0) roo (
(1 _ (Jo)2
1-00
logy,
y
1)
1 + yeV e
-Av 2 /2
dv
(2ZI-0'0)
(1 - (Jo)2 .
The error terms RI and R2 are clearly of order O(logyjy) (assuming, as we
may, that e ::; 2), and the same holds for the term R 3 , since 2 = log z «
logy, and by Proposition 3.1,
This proves (8.4).
Lemma 8.3. We have
1
r(y,z;A)V(z)=-2'
1tl
(8.5)
10'0+i F(s) y s-I k(s-l)ds
O'o-i
s
+ O(yO'o-1 e(l-e/3; 1-0'0 (1og A)4 ).
Proof. In view of the previous lemma, and since
logy
ys/2-1
yO'o-1
-y- « -2 - <
-- 2 '
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OSCILLATION THEOREMS
59
it suffices to show that the contribution of the range 1 ::; It I ::; To to the integral
in (8.4) is bounded by the error term in (8.5). We split this range into the two
parts 1 ::; It I ::; Tl and TI < It I ::; To, where
TI = min(To' exp(2'3/2-e)).
Since
(8.6)
Ik(O'o - 1 + it) I = exp (
(0: - 1)2 - t 2 )
/2/ 2).
0 2A.
«e-
and
(8.7)
T~-O'qog(1 + To)2' exp ((12ZI-;~)
- 0'0
max IF(O'o + it) I «
It l:5 To
«
To2' exp(O(u)) «l/e
by Lemma 7.3 and Proposition 3.1, the integral over the range TI < It I ::; To is
bounded by
«
max IF(O'o + it)1 [To Ik(O'o - 1 + it)1 dt
It l:5 To
«l/e
iT!
[00 e-t2/2Adt«l/eVie-T(/2A.
iT!
If A ::; To, then we have Tl
above bound becomes
~
A by (8.2) and the definition of T 1 , and the
«l/eVie- Td2 «l/ee- z / 2 « l/y,
since TI ~ min(y, exp(2')) = z and z» (logy)1/(1-e/2) in the range (Re) . On
the other hand, if A> To, then (8.2) implies exp(2'2/3-e) > To, in which case
TI = To and the range TI < It I ::; To is empty. In either case, the contribution
of this range is of the desired order, and to prove (8.5) it remains to show that
the same is true for the range 1 ::; It I ::; Tl .
Since F(s) = F(s) and k(s) = k(s), it suffices to consider the integral
1=
1
0'0+iT! F(S)yS-1
Now
k(s-1)ds.
. 0'0- 11
1 = IY
I'
where
with
(8.8)
s
O'o+i
G(t) =
+ it)
k(O'o 0'0 + It
F(O'
o.
1 + it).
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J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
60
Hence, in order to prove the desired estimate, it suffices to show that
1 1-0:o(1og).) 4 exp {
1111 «£').
(8.9)
(
1 - '6))
3 u }.
At this point a trivial bound of II by the integral over the modulus of G(t)
would not be sufficient to yield (8.9). Therefore, we have to proceed more
carefully, and we first apply an integration by parts to get
II
yit
ITI = -'l-G(t)
lOgy
I
which implies
1111«
1
-'1lOgy
!TI o'(t)ylt.dt,
I
~L!!}~~IIG(t)I+ lT1Io'(t)ldt}.
We estimate the terms on the right-hand side using Lemmas 7.4 and 7.5.
By Lemma 7.4 (with 6/3 in place of 6) and Proposition 3.1 we have, for
1~t~Ti'
.
1-0:
2
{
IF(O'o + It)l<< t 010g (1 + t)exp
(
« t 1-0:0log2 (1 + t) exp {
(
1+
Z 1-0'0 }
'6)
3 ~
e'u ( 1 + 0 (log
1u) ) }
1 + '6)
3 £'
« tl-O'o 10l(1 + t) exp { (1 -
i) u} ,
where in the last estimate we have used the identity e'u = u~u + 1 , the estimate
(3.2) for ~u' and the bound logu ~ (1 - 6)£' . It follows that
IF((1o + it)1 _t 2 /2).,
e
t
«exp{(1 - 6/3)u}.
max IG(t)l« max
t=I,T1
t=I,T1
A similar argument shows that, for 1 ~ t ~ TI '
.
1-0:
3
IF , (0'0 + It)1
« t 0 log (1 + t) exp{(1 - 6/3)u}
and hence,
. )1 _t2 /2).,
IF(
')1 _t2 /2)., ( !!... )
F '(
0'0 + It e
lo'(t)1 « I 0'0 + tIt e
+
t
t+).
-0:
«t 0(1 + t/).)e
-flU
3
log (1 + t) exp{(1 - 6/3)u}.
It follows that
dt (1og).) 3exp {( 1 - '6)}
3 U
! TI 10' (t)1 dt « {!).,dtto'o + I1 1)"foo te 2/2).,}
I
-t
I
«).
1-0:
o(log).)
4
exp{(1 - 6/3)u}.
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OSCILLATION THEOREMS
61
Combining these estimates we obtain (8.9).
Lemma 8.4. With c5 = 1I.:? we have
/to+o F(a
1
+ it)yGo-l+it
k(ao -l+it)dt
ao + It
o.
r(y,z;A)V(z)=-Re
to-o
+ O(yGo-l e(I-B/3;I-GO(10gA)4)
'It
(8.10)
+
o(yGO-~(so)1 exp ( -
410:2 u)).
Proof. Since F(ao - it) = F(ao + it) and k(ao - 1 - it) = k(ao - 1 + it), the
integrand in (8.10) changes into its complex conjugate when t is replaced by
-t. Thus, the main term on the right-hand side of (8.10) can be written as
1
-.
2m
(l
GO - i(to-O)
Go-i(to+O)
+
l
Go +i(to+O) ) F(S)yS-1
k(s-l)ds.
Go+i(to-o)
S
By Lemma 8.3, then, it suffices to show that the contribution of the ranges
It I :::; to - c5 and to + c5 :::; It I :::; 1 to the integral in (8.5) is of the order of the
second error term in (8.10).
For the range It I :::; to - c5 the trivial estimate
ro+i(to-O) F(S)yS-1 k(s - 1)1 «yGo- \ max IF(ao + it)1
S
Itlsto-o
1} Go-i(to-o)
is sufficient, since to« II.:? by Proposition 3.1, and
max IF(ao + it)l« !p(so)lexp(-uI310g 2 u)
ItlSto-o
by Lemma 7.5. The second range to + c5 :::; It I :::; 1 requires a more delicate
argument. It suffices to consider the integral
1= /
1 F(
10
+0
= yGo-1
.) Go-I+it
ao+lt~
/1
10+0
ao + It
k(ao -l+it)dt
G(t)/I dt = yGo-I II '
say, where G(t) is defined by (8.8). We have to show that
!P(so)1
( 1111« ~exp
(8.11 )
oZ
Partial integration gives
II
«
-11_ { max IG(t)1 +
ogy to+oStsl
u)
2
410g u
1/1
10
+0
•
G' (t)/I dtl}.
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62
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
By (S.S), (S.6), and Lemma 7.5, we have
(S.12)
Thus, in order to prove (S.ll), it remains to show that the integral
12 =
/
1
10+0
it
,
G (t)y dt
satisfies
2
(S.13)
1121 « IF(so)1 exp( -u/410g u).
By (S.S), (S.6), Lemma 5.1, and Proposition 3.1, we have, for It I :::; 1,
(S.14)
0'( ) = iF((Jo + it)k((Jo - 1 + it) {F'((Jo + it) _ _1_ k'((Jo - 1 + it)}
t
(Jo + it
F((Jo + it)
(Jo + it + k((Jo - 1 + it)
= iG(t){¢'((Jo
.
= IG(t)
+ it) + 0(1)}
{Zl-Uo-it
1 _ (Jo _ it
= GI (t)z
-il
(Zl-UO
)
+ 0 (1 _ (Jo)2 + 0(1)
}
+ O(uIG(t)I) ,
where
iz l - uo
GI(t) = 1- (Jo _ itG(t).
In view of the bound (S.12), the contribution of the O-term to the integral 12
is of order
2
«u max IG(t)l« uIF(so)1 exp( -u/310g u),
10+0::;t~1
which is sufficient. Moreover, a further integration by parts yields
(S.15)
1/1 G,(t)Z-il/1dtl«1
10+0
Since
/ / ){
og Y z
max IGI(t)I+/1
10+0::;t~ I
10+0
log(y/z) = (u - 1)2» u2
if u 2: 2 , as we may assume, and since
Zl-uo
ulogu
IGI (t)1 « 11 _ (Jo _ itIIG(t)1 « -ltl-IG(t)l,
,
Zl-uo
. 2IG(t)1 + 11
. IIG (t)1
11 - (Jo - ltl
- (Jo - It
Zl-uo
Zl-uo
« -IG(t)1 + -(IGI(t)1 + ulG(t)l)
tit
,
IG I (t)1 «
«u
Zl-uo
2
2
102g
t
uIG(t)1
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IG~(t)ldt}.
OSCILLATION THEOREMS
63
by (8.14) and Proposition 3.1, the right-hand side of (8.15) is bounded by
«0;)1
..z;
1.
2
1
u log
U
u2 log
max IG( t ) I{ -+
2 U dt }
to+t59$1
to
to +15
t
u2 log2 u
£'
max IG(t)1
to
to+t5 9$1
«
«
u2 (log u)2 IF (so)1 exp( -u/3log2 u),
where the last estimate follows from (8.12) and the bound to » 1/£' (cf.
(3.12)). This implies the desired bound (8.13) and completes the proof of
Lemma 8.4.
Lemma 8.5. With
a = 1/£'
we have
to +15 F(a + it)yUo-l+it
o
.
k(ao - 1 + it)dt
to-t5
ao + It
1.
(8.16)
=
(1 + 0(_1_)) ySo-IF(so)k(so - 1).
log u
so£' y'u/2n
Proof. The integrand in (8.16) can be written as
(8.17)
1 )) ySo-1 k(s - 1)
( 1+0 ( £'
soo
exp{i(t-to)logY+QJ(ao+it)},
since t = to+O(l/£') = 0(1/£') and k(ao-1+it) = k(so-1)(1+0(1/£')) in
the range of integration. Expanding QJ(ao + it) in a Taylor series about t = to'
we obtain
i(t - to) logy + QJ(ao + it) = QJ(so) + i(t - to)(logy + QJ' (so))
• )
- '1
2 (t - to )2"()
QJ So - 3ii ( -r - to )3 QJ III ( ao + l-r
with a suitable -r = -r(t), l-r - tol :$ It - tol. By the definition of so' the linear
term here vanishes. Moreover, from Lemma 5.1 and Proposition 3.1, we see
that
"
_Zl-So£, (
(
1
))
QJ (so) = 1 _ So
1 + 0 (1 - ao)£'
2
= u£' (1 + O(ljlogu))
and
ZI-uo£,2 (
(
1 )) =u£'3( 1+0 (1))
logu .
1-ao 1+0 (i-ao)£'
Thus we can write (8.17) as
III
•
IQJ (ao+l-r)I::;
(8.18)
(1 + O(l/£'))M exp{ -t(t - to)2u2'2 + R(t - to)}'
where
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64
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
and the function R(t) satisfies
Setting
_ (IOgU)I/2 -_ -1 (IOgU)I/2
--
6 -6 1
.!?
U
u
'
we have, for It - tol :::; 61' R(t - to) = 0(1), and
exp(R(t - to))
and for It - tol :::; 6
= 1 + O(R(t -
to))
= 1 + O((t -
to)2.!?210;U) '
= II.!? ,
where the last estimate follows since
rOO 2/2
1- e- r dr == v'2n
00
and
rOO e-r2/4 dr = /00 e- r2 / 4 dr« e-(logu)/4 «
1t5 vu£'
y'logu
l
This proves (8.16).
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_1_.
log u
65
OSCILLATION THEOREMS
Proof of Theorem C. We now suppose that A satisfies the hypothesis log u :::;
A:::; exp(2'3/2-e) of the theorem. Combining Lemmas 8.4 and 8.5 we obtain
o(
(8.19)
r(y z· A) _ Re /0- 1F(so)k(so - 1) +
yO"o-llF(so)1 )
, , nV(z)so2' y'u/2n
V(z)2' y'Ulogu
yO"o-1 e(l-e/3)u A1-0"0 (IOgA)4)
(
+0
V(z)2'
+0
Since
k(s - 1)
o
( yO"o-llF(so)1
V(z)2'
exp
(
u)) .
- 410g2 u
= e(So-I//2}. = 1 + o(!) = 1 + 0(_1_)
A
logu
by the assumption A 2: log u of Theorem C, and since by Mertens' formula,
the main term in (8.19) is equal to
Re
y~o-IF(So)
ne y soy'u/2n
+o(yO"o-IIF(So)l) =Rep(y, z)+o(IP(Y, Z)I).
y'Ulog u
log u
The same argument shows that the first and third error terms are bounded by
O(lp(y, z)l/ log u). Finally, using the seconds part of Lemma 7.5, the second
error term is seen to be of order
«yO"o-1 e(l-e/3)u A1-0"0(logA)4
= Y 0"0-I IF (so) 1exp
{e
1 ) )} A1-0"°(lOgA) 4
- 3u (1 + 0 (logu
« Ip(y, z)le -eu/4 A1-0"0 (lOgA{
Altogether, we obtain
r(y, z; A) = Re p(y, z) +
o( 'Pi~~ :)1) + O(lp(y, z)le -eu/4AI-O"O(logA)\
This implies the desired estimate (3.11) in the case A satisfies, in addition to
the hypotheses of the theorem, the inequalities A1-0"0 :::; eeu/8 and log A :::; u2 .
Therefore it remains to show that the same estimate holds for values A in the
range
(8.20)
. (2
eu 0"0) )
mm u , 8(1 _
:::;
1 1
c£l3/2-e .
og/\.:::;..z;
Since 1 - 0"0 = (~u/2')(l + 0(1/log2 u)) «logu/2' by Lemma 5.1, the range
(8.20) is only nonempty in the case u «2' , which we may henceforth assume.
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J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
66
We shall show that
(8.21)
+o(~)
r(y, Z;A) =r(y, Z)+O()x)
holds for any A ~ 1. Now if A is in the range (8.20), then the error terms in
(8.21) are bounded by
1
Z
v'J. + Y «exp
{
eU}
+e
- 16( 1 _ 0'0)
_u 2 /2
+e
-(u-I)..2"
~~~ ( 1 + 0 Co~ u) )}+ e_u /2 + e-(u-I)..2"
« exp { -
2
« exp( _U 3/ 2) « Ip(y, z)1
logu
in view of Propositions 3.2 and 3.2, and the bound u«.!? Thus, setting
Al
= exp { min ( i,
8(
/~ 0'0)) } ,
we obtain
r(y z'A)=r(y Z'A )+o(IP(Y,Z)I)
"
"
logu
I
+ o(IP(Y,
= Rep(y z)
,
Z)I)
logu
since (3.11) holds for A = AI' This shows that the desired estimate (3.11)
remains valid in the range (8.20).
To prove (8.21), we note that by sieve methods we have, for Y2 ~ YI > 0,
cl>(Y2' z) - cl>(YI ' z)
It follows that for
Iv I :::;
«
(Y2 - YI
+ z) V(z).
1,
v
Ir(ye ,z) - r(y, z)1 =
V ,z)
Icl>(ye
yeVV(z) -
cl>(ye V, z)
:::; yeV V(z) 11
«
(ye V + z)lvl
v
ye
z
cl>(y, z)
yV(z)
-
I
1
v
1+ yV(z) Icl>(ye
v - 11 + z
+ Yle
'-'------'-e
y
« Ivl +-.
y
Since for any y' > 0 ,
,
cl>(y' , z)
Ir(y , z)1 :::; 1 + y'V(z) «1
z
+ y'
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,
v
,z) - cl>(y, z)1
67
OSCILLATION THEOREMS
we obtain
9.
PROOF OF THEOREM
Bl
We fix a (small) positive number e, real numbers y and z in the range R e ,
and a positive integer q. As before, we define u = logy I log z and £' = log z .
Throughout this and the next section, the implied constants depend at most on
e , and thus, in particular, are independent of k.
Since the function rk and the hypothesis (3.5) of Theorem Bl depend only
on the prime divisors of k that are < z , we may assume that k has no prime
factors ~ z. Thus (3.5) reduces to
L 1 :5 2 log£'.
(9.1)
plk
We begin with the following simple lemma:
Lemma 9.1. Let "f.,' denote summation over integers all of whose prime factors
divide k. We have, for any a: ~ el2 and D ~ 1,
L
, 1
d?D
Proof. We have
, 1
L d o. :5 d?1
L
d?D
dO.
«
£,1/2
Do.(l-e/4) •
, 1 (d)o.(l-e/4)
dO.
D
= D- o. (l-e/4) ~'_1_
L- do.e/4
d?1
= D- o. (l-e/4) IT
plk
(1 __1_)-1
P Ol.e/4
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68
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
Now if v is the number of prime factors of k and PII denotes the vth prime,
then
g(1-
1)
p ae / 4
-I
(9.2)
:::;
=
1- p ae1)
/
XI 1- pe1)-1
exp {o( 2:: e~/8)} exp{O(p~-e2/8)}
(
p~I
4
p5, Pv
-I
(
2 /8
:::;
=
p
«exp(vl-e2/IO) « 2 1/ 2 ,
since p II « v log v and v « log 2
by (9.1). This proves the asserted estimate.
Next, we define a smoothed form rk(y, z; A) of rk , analogous to the function r(y, z ; A) in Theorem C, by
rk(y, z; A) =
..;x
f'C
1
00
V
rk(ye,
z)e -Av
2 /2
dv.
v2n -00
The following lemma relates the functions rk(y, z; A) and r(y, z; A). Here,
and in the sequel, r.' has the same meaning as in Lemma 9.1.
Lemma 9.2. For any A 2: 1 we have
(9.3)
rk(Y'Z;A)=rr(1-~){ 2::' ~r(~,z;A)+O(
plk
d5,yI-t/2
Y
1!4e/5)}'
Proof. We start out with the identity
<l>k(y' ,
z) = 2::' <I>(~ , z) ,
(y' > 0),
d~1
which follows easily from the definition of <l>k (y, z). By Lemma 9.1 the contribution of the range d > y l-e/2 to the sum is bounded by
CLJI/2
'£JI/2
"",' Y ,
,..z, ..z;
:::; L...J d « y (l-e/2)(I-e/4)« y ---=-1---=3,--e/,..,-4·
d>yl-,/2
Y
Y
21/2)
+ 0 ( y' yl-3e/4
'
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69
OSCILLATION THEOREMS
where V(z) =
I1p <z(l -
lip) and, in the last estimate, we used the bound
, ( 1) -I + 0 ( Y, .;?1/2
)
2:'y'd = Y, d~1
2:'1(j + 0 ( 2:'Y')
d = Y II 1 - Ii
1-38/4 .
plk
Y
d'5,yl-./2
d>yl-./2
It follows that
r ( , z)
k Y ,
(1 - P!) cl>k(Y'
, z) - 1
y'V(z)
II (1 - -1) { 2:'1(jr (y'd' z ) + 0 (
= II
plk
=
P
plk
.;?1/2
) }
1-38/4·
V(z)y
d-:;,yI-e/2
Letting Y' = ye V and integrating both sides of this identity against the weight
function J)..127C exp( _)..v 2 12) , we obtain
rk(y, z; )..)
= II
plk
(1 - !) {
P
2:' ~r(~, z; )..) + o( V(z)y
.;?1:~38/4)}'
d<yl-./2
which in view of the bound
implies the asserted estimate (9.3).
Lemma 9.3. We have
and
(9.5)
where
(9.6)
Ck(y, z)
= II
plk
(1-!) (1P
P
I_~ /2)-1.
u
Proof. By the estimate (3.15) of Proposition 3.2 (with el2 in place of e), we
have for d ::::; yl-e/2 ,
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J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
70
Hence the left-hand side of (9.4) equals
,
1
(9.7)
p(y, Z)
d l -C../2'+O(I/2')
L
d$.yl-./2
= p(y,
1
)
L ' d l - 1eul2' + 0 (L' d l - e./2'+O(I/2')
z) {
d~1
d~z
1 ~'
+ 0 ( 2' L..J
d<z d
logd )}
1-c../2'
.
The main term inside the braces can be written as
'1
(
L
d l - e./2' = II 1 d~1
plk
1 ) -I
P
= ck(Y,
l-e./2'
z)
(1)-1
II
1- P
plk
by the definition of ck(Y, z). Further, since 1 - ~u/2' ~ 2e/3 in the range
Re (assuming that the constants Uo and Zo in the definition of R8 have been
chosen sufficiently large), from Lemma 9.1 we get
L
'1
d~z
dl-e./2'+O(I/2')
'1
2'1/2
1
«L d8/2 « z(8/2)(1-8/4) « 2'1/2
d~z
and
1 ~'logd
2' L..J dl-e./2'
d<z
«
1 ~ logp~' 1
p8/2 L..J d 8/ 2
2' L..J
plk
1 ( L 1)
«2'
plk
d~1
( 1II
plk
P
1 )-1 « --rJ3
1
8/2
2'
in view of (9.1) and (9.2). Therefore, both error terms in (9.7) are bounded by
0(2'-1/3), and the right-hand side of (9.7) becomes
ply, z)CklY,
z)!J (1- ~) (1 + 0C2}1/')).
-I
This proves (9.4); the second estimate (9.5) can be established in the same way.
For the remainder of this section we restrict (y, z) as in the statement of
Theorem B1 to the range (R I / 6+B)' This restriction would not be necessary if
the Riemann Hypothesis were assumed.
Lemma 9.4. There exist numbers z± in the range
. ( z 1-2/(u+2) -z
I )
mm
'2
(9.8)
<
- z ±<
- z
such that
(9.9)
{
Re p(y, z+) ~ exp{ -u~u - c i u},
Re p(y, z_) ~ - exp{ -u~u - c i u}
holds, where c i is a suitable positive constant depending at most on e.
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OSCILLATION THEOREMS
71
Proof. By Proposition 3.2 and (3.2) we have, for any z' in the range (9.8),
Ip(y, z')1 = exp{ -u' ~UI
where u'
=
log y / log z' . Since
u'
= u log z, = u (1 + 0
logz
+ O(u'n,
(.!u + -1
1 ))
og z
for min(z/2, ZI-2/(U+2)) :::; z' :::; z, we may replace u' by u in this estimate.
To prove (9.9), it therefore suffices to show that there exist values z± satisfying
(9.8) such that
(9.10)
1Argp(y, z+)1 :::; n/3,
1Argp(y, z_) - nl :::; n/3,
where Arg denotes the principal value of the argument.
First we consider the case when u:::; U. By the estimate (3.15) of Proposition 3.2 we have, for u:::; u' :::; u + 2 ,
I
+ n(u' - u) + O(I/10gu).
u + 2] such that
argp(zu , z) = argp(zu, z)
Hence there exist numbers u± in [u,
(9.11)
provided u 2': Uo with a sufficiently large constant uo ' as we may assume.
Setting z± = zu/u± , we then have
< z± <
Zu/(u+2) _
_ z,
so that (9.8) is satisfied, and
= argp(z~± , z±) = argp(zu± , z) + O(u/2')
argp(y, z±)
by the last estimate of Proposition 3.2. In view of (9.11) and our assumption
u:::; U, this implies (9.10).
This argument would go through for u :::; Z5/12, if we would use Huxley's
prime number theorem and the estimate (3.16) of Proposition 3.1, but it would
break down for larger values of u. For the range U < u :::; z5/6-e , therefore,
we use a somewhat different approach that depends on an "almost all" type
estimate for primes in short intervals. Let
h=zU/u,
so that, for 1 :::;
assumption that
holds with z± =
First we note
3.2 satisfies
zn=z-nh
(l:::;n:::;n o =[z/2h]),
n :::; no' z/2:::; zn :::; z, and zn - zn+1 = h » zl/6+e by our
(y, z) lies in the range (R 1/ 6+e)' We shall show that (9.9)
zn for suitable indices n± :::; no'
±
that the error term function E(z, z) defined in Proposition
l
z
z/2
IE(t, t+h)ldt«A
zh
(log z)
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A
72
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
for any fixed constant A. This follows from Selberg's work [Se] and Huxley's
zero-density estimate [Hu] in view of the bound h» zl/6+e; for the argument
see Lemma 2 of Nicolas [Ni]. A simple consequence of this estimate is that
(9.12)
IE(zn' zn+I)1 «hl2'
2
holds for all n :5 no with at most 0(no/2'2) exceptions. If Z is large enough,
this implies that there exist at least 2' consecutive indices n :5 no for which
(9.12) holds, say n = n I + 1 , ... , n I + J , wher J = [2'] and 1 :5 n l :5 no - J .
Let
Uj
By the definition of zn and h
Uj+1 -
uj
= log(zn
-
1
= logyllogzn,+i'
we have, for j = 1, ... , J,
logy
_ (j + l)h)
logy
log(zn - jh)
1
u2'log((zn - jh)/(zn - (j + l)h))
1
(logzn )2
-
1
1
+ 0(2')
hu
1
x-=-z2'
n'
Thus, in view of (3.15), the argument of p(z~j, Zn ) increases in steps of size
1
'I
x lin, as j ranges over the integers in the interval [1, 2'}. It follows
that, for suitable indices j± :5 J, we have (9.11) with zn in place of Z and
1
u± = uh . Using the esimate (9.12) for n = n l + j, 1:5 j :5 J, Proposition
3.2, and the definitions of hand J, we see that with z±
have
argp(y, z±)
= zn +j'± = yl/U± , we
1
= argp(z~\j' ± ,zn +j'± )
1
1
(
= argp(z~~, zn l ) + 0 ( ~ nl+J
~ IE(zn'
n-nl
zn+I)1
h))
+ 2'2 + ze
l( Z
U ( 2'2
h+
eZ) )
= argp(zn~ , zn) + O\J
U
= argp(z~~, Zn l ) +
O( k ).
This estimate, in conjunction with (9.11) (with zn in place of z), implies
1
(9.10), and thus completes the proof of Lemma 9.4.
ProofofTheorem Bl. From Lemma 9.2, Theorem C, and Lemma 9.3 we obtain,
for log U :5 A :5 exp(2'3/2-e) ,
rk(y, z; A) = ck(y, Z){ Rep(y, z)
+ 0CPi~~:)') + 0CP~;/:)I) }
+ 0(yll4E/5 ).
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73
OSCILLATION THEOREMS
The last error term here can be estimated, using the bound u ~ z5/6-e and
Proposition 3.2, by
Y -1+4e/5
since ck(Y, z)
rk(y, z; J.)
~
«
«
exp{ -( 1 - ~e )u2'}
Ip(y, z)1
logu
«C
« exp{ -( 1 + leO )u~u}
(y z) Ip(y, z)1
logu'
k'
1 . It follows that
= ck(Y,
Z){ Rep(y, z) + o('Pi~~:)') + o('P~;/:)I)}.
We apply this estimate with YI = Y - y/2logy in place of y, and z± in place
of z, where the numbers z± are chosen according to Lemma 9.4 so that (9.9)
(with y replaced by y I) is satisfied. Adjusting the constant CI if necessary, we
obtain
(9.13)
Note that the numbers z± here depend only on y and z, but not on k or J..
We now set ~ = 1/2logy, A = log(1 +~), so that
a
y~YI(1+~)=Yle ,
(9.14)
y(1Yl e- a ,
lO~Y) ~ YI(1-~) ~
= (logy)4 = (U)-4. With this choice of parameters we have
VI {a
v
-.l.v 2 /2
(. f"7i = roo -.l.v 2 /2
)
z+; J.) = v'2i J-a rk(YI e ,z+)e
dv + 0 v A..z; Ja e
dv
and choose J.
rk(YI'
VI fa
= .I'C
v
-Av 2 /2
rk(Yl e , z+)e
dv
v27l' -a
~ max rk(Ylev , z+)
ivi::5a
~
max
y(I-ljlogy)::::;Y' ::5y
+ O(.!£'e -Aa /2 )
2
+ O(1/y)
rk(y', z+)
+ O(1/y),
and a similar estimate holds with z _ in place of z +. Combining these estimates
with (9.14) yields
max
.v(I-ljlogy)::5y' ::::;y
±rk(y', z±) ~ exp{-u<!u - CIU}
+ O(1/y) .
In view of the hypothesis u ~ z5/6-e , the error term here is of smaller order than
the main term. Therefore we obtain the desired estimates (3.7) with suitable
numbers y± satisfying (3.6), after adjusting the constant c i if necessary.
10. PROOF OF THEOREM B3
Since the proof is to a large degree analogous to that of Theorem B 1, we keep
the exposition short. We fix e > 0, real numbers y, z in the range R e , and
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74
1. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
a prime p ~ z. We concentrate on the case when p
p < z is similar and simpler.
We write
Cl>(p) ( z)
y,
~
z; the argument for
= ",(P) ~
rp(n) ,
L...J
n:;;x
where E(P) denotes summation over integers n that are not divisible by p or
by any prime < z ,
and
p:
c = C(P)(z) =
(1
p ;?::z
1))'
+ p'(p,l_
p'>Jp
We recall that r(P) and ,(P)(y, z) are defined by
Cl>(P)(y, z) = yCV(P)(z)(l
+ r(P)(y,
z))
and
For A > 0 we set
r
(P)
vA
(y, z; A) =..j2i
,(P)(y, z; A) =
r
ly/2
1
00
-00
r
(p)
r(P)(t, z;
v
(ye ,z)e
_;,v 2 /2
dv,
A)~t.
The following lemma is analogous to Lemma 9.2.
Lemma 10.1. For any A ~ 1 we have
(10.1 )
r
(P)
(y, z; A)
1
= C(1 _
1/)
p
{
",(p)
L...J
d:;;yl-e/2
-
,i(d)
d (d) r
rp
(Yd' z; A)
(Y .) }+ 0 (.:?)
yl-e/2 '
1 ",(p) ,i(d)
drp(d) r dp' z, A
~:;;7-e/2
where the summation in E(P) is restricted to integers d that are not divisible by
p or by any prime < z .
Proof. Using the identity
(n
~
1),
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OSCILLATION THEOREMS
75
we obtain, for any y' > 0,
<I>(p) ( ,
y,
) - ~(P) ~ ,i(d) _ ~(p) f./(d) ~(p) 1
z - L...J L...J-(d) - ~ -(d) ~
n~y' din rp
d~l rp
n9'/d
~ ~t ~2(~i (~(~, z) -~(:»)
:F'~: ~(~i~ V(Z){ \ - ~ + r(~, z) - H;~, z)} + O(i:/2)
-Y
'(p)
= Cy V
~(p) J.l2(d) (y'
1
{
(z) 1 + C(l _ 11) ~
d (d) r
P d9 1-,/2 rp
~(p)
1
- C(l - 1/p)p
~(p) J.l2(d)
~ rp(d)d
d~l
J.l2(d) (y'
dp'
d~-'/2 drp(d/
where
C=
d'
z
)
Z
) }
(y')
+ 0 yl-B/2 '
IT (1 + p'(p'1)
_ 1)
=,
p ~z
p'~p
is defined as above and the error term absorbs the contribution of the range
d > yl-B/2 to the sum over fJ>(y'ld, z) and that of the range d > yl-B/2 to
the sum over fJ>(y' I d p, z). It follows that
r
(P)
,
1
{ ~(p) J.l2(d) (y'
)
(y, z) = C(1 _ lip) ~
drp(d) r d' z
d~yl-'/2
1
~(p)
/(d)
- Pd~-'/2 drp(d) r
(y'dp' )} + 0 (:f)
yl-B/2 .
z
Integrating both sides with y' = ye V against the weight function VA/2n'
. exp( -AV 2 12) , we obtain (10.1).
Lemma 10.2. We have
and
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J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
76
Proof. This result can be proved in the same way as Lemma 9.2, using the
estimates
~(P)
Jj2(d)
d U ~(d)
L..J
d?l
= exp
{( ~
1
)}
0 L..J p'u (p' _ 1)
p'?z
= 1 + O(l/v'Z)
(0' ~ e/2)
and
We omit the details.
ProofofTheorem B3. From Lemma 10.1, Theorem C, Lemma 10.2, and Proposition 3.1, we deduce that, for logu :::; A:::; exp(2'3/2-e),
r(P)(y, Z;A)
It follows that
y(P)(y, z; A)
=
i
=
Re
= Rep(y,
z)+O(lp(y, z)l/logu).
d '
r(p)(y', z; A)~
y/2
Y
y
i
y
y/2
dy'
p(y , ,z)-,
y
+0
(1ogu i
-1-
Y
y/2
d Y') .
Ip(y , ,z)I-,
Y
Since by Proposition 3.2,
p(y', z)
we have
Re
= p(y,
z)
(~) -eul.? (1 + oCo~u))
(y/2:::;y':::;y),
rly/2 p(y', Z)d~'
= 2eu /.? -1 Rep(y, z) + o(IP(Y, Z)I).
y
~u/2'
logu
Replacing y by Yl
=Y -
y/2logy and applying Lemma 9.4, we deduce that
±y(P)(Yl' z±; A)
~ exp{ -u~u - c1 u}
holds with suitable numbers z± satisfying (3.4) and an appropriate constant
c1 • Taking A = (logy)4, we complete the proof of Theorem B3 by following
the argument at the end of the preceding section.
III.
OSCILLATION THEOREMS FOR PRIMES
11.
PROOF OF THEOREM
Al
For the proof of Theorem Al (as well as that of Theorem A2), we require
the following result that refines Proposition 2 of [HM]. It is here that we appeal
to the Linnik-Gallagher prime number theorem.
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OSCILLATION THEOREMS
77
Proposition 11.1. Fix e > o. For any squarefree integer n > 1. all of whose
prime factors are ::; n I-e. there exists a divisor P of n. with niP prime. such
that if (a, P) = 1. x ~ p2 • and x ~ h ~ x I exp( Vlog x). then
(1.11) O(x + h ; P, a) - O(x; P, a)
= ¢>~) {I + O(e -c1ogx/logP + e -c..jlogx)} ,
where c is a constant depending only on e.
Proof. For a given value c > 0, a character X (mod q) is called" c-exceptional"
if L(P, X) = 0 for some real P> l-cflogq. It is well known (cf. [Da, p. 94])
that there exists a constant Co > 0 for which there is at most one co-exceptional
character X (mod q) for each q ~ 2. First we show that there is a divisor P
of n, where niP is prime, for which there are no eco-exceptional characters
X (mod P).
Suppose that this is false and let ql be a divisor of n with nlql prime. Let
Xl be the eco-exceptional character (mod ql) and let p be a prime divisor
of the conductor of Xl. Set q2 = nip; note that p is a divisor of ql and
so of n. Let X2 be the eco-exceptional character (mod q2)' and denote the
characters of modulus n induced by Xl and X2 as X3 and X4' respectively.
Since p divides the conductor of X3 but not that of X4' X3 and X4 cannot be
the same character. However, since by our assumption L(Pi , Xi) = 0 for some
numbers Pi > 1 - eco/logqi and i = 1, 2, we have L(Pi' Xi+2) = 0 where
Pi> l- ecoflogqi ~ l- coflog n as each qi ~ ne • Thus X3 and X4 are distinct
co-exceptional characters (mod n). This, as was already pointed out, cannot
happen.
Choosing P as above we have, under the hypotheses of the proposition,
O(x + h; P, a) - O(x; P, a)
=
1
cp
(P)
L
x(a)
X (mod P)
L
X(p) logp
x<p5,x+h
and therefore,
IO(x + h; P, a) - O(x; P, a) - hlcp(P)1
::;¢>~
()
{I
L
x<p5,x+h
10gp-hl+L
kiP
k>l
L*
X (mod k)
IL
x<p5,x+h
X(p)IOgpl} ,
where in E* the summation is restricted to primitive characters. By the choice
of P, the characters appearing in the last sum all satisfy L(P, X) :f. 0 for
P > 1 - eCol log P. Thus, estimating the sum over p by a sharp form of the
prime number theorem, and the character sum by Theorem 7 of Gallagher [Gal
(with Q = max{P, exp( Vlog x )}) , we obtain
IO( x + h ·,P, a ) -
)
h I
h {-clogx/logp
x, P, a - cp(P) «cp(P) e
+ e -c..jlogx} .
0(·
This proves (11. 1).
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J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
78
Pro%/Theorem AI. We give the proof of (2.3), the proof of (2.4) being almost
identical. Given q and x satisfying (2.1) and (2.2), we let Y = x/q and define
z = log x/log Y so that (y, z) lies in the range (R 1/6+8) of Theorem B 1 (with
Uo = 1 + e) provided that e is sufficiently small and %(e) sufficiently large.
We next take z+ as in Theorem Bl, define n = TIp<z+ ,P'rqP, and choose P as
in Proposition 11.1. Finally we set k = q n/ P , so that
(11.2)
P
=
IT p.
p<z+
p'rk
The hypothesis (2.1) on q ensures that k satisfies the condition (3.5) of Theorem Bl for q ~ qo(e) , since k = qn/P has only one prime divisor more than
q and we have
logx
logq
Z=
«log2
- -q
log(x/q)
in the range (2.2). Thus by Theorem Bl, there exists a number Y+ in the range
(3.6) for which the first inequality in (3.7) holds. Since
<l>k(Y+' z+) =Y+
IT
p<z+
p'rk
(1- P.!.)(1
+rk(y+, z+)) =Y+
rp~)(1 +rk(y+, z+))
by (11.2), this inequality can be restated as
rp(P)
(11.3)
<l>k(Y+' z+) ~ Y+ ---p-(1 + exp( -u~u - c1u)),
where u = logy / log z .
Consider now the matrix L = (a rs ) , where
= {IOg(rp+qs)
ifrP+qs prime,
0
otherwise,
and where rand s run over the values r '" R (that is R < r :::; 2R) and
1 :::; s:::; Y+ with
a
rs
( 11.4)
R=
2~ exp( -y'logx).
Let IMI denote the sum of the entries of L . For given s, the sum of entries
in the sth column is
O(2RP + qs; P, qs) - O(RP + qs; P, qs).
This vanishes if (qs, P) > 1, and for those s with (qs, P) = 1 it may be
estimated by applying (11.1) with x, h, and a replaced by P R, P R , and qs,
respectively. By (11.2) and the definition of <l>k' there are precisely <l>k(Y+ ' z+)
values of s satisfying (s, P) = 1, or equivalently (since (q, P) = 1), (qs, P) =
1 . Thus, we obtain
(11.5)
IMI = (~(Pp) (1 + O(e -clogRPflogP) + O(e -C"'IOgRP))<I>k(Y+ '
'I'
~ Ry +{I + expC-u<!u - C~ un
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z+)
79
OSCILLATION THEOREMS
with a suitable constant c~ , since
10gRP x logx ~ logx = logy = ulogz
10gP
z+
z
and
Vlog RP ~ Vlog x ~ log y = u log z
by (11.4) and the definitions of z+ and z.
On the other hand, the sum of the entries in row r is
e(rP
+ qy+;
q, rP) - e(rP; q, rP)
= e(rP + qy+; q, rP),
since rP:5 2RP < q by (11.4) and (2.2), and rP is not prime. This vanishes
unless (r, q) = 1. Thus,
IMI =
(11.6)
L
r",R
e(rP + qy+; q, rP).
(r,q)=1
The number of r satisfying r",R and (r, q) = 1 isjust Rrp(q)/q+O(r(q)) ,
and, since R > XI/2 , the error is negligible. Thus, by (11.3), (11.5), and (11.6),
there is at least one r for which
.
qy+
e(rP + qy+, q, rP) ~ rp(q) {I
+ exp(-uc;u - c1 un,
II
where C~' is an appropriate constant depending at most on e. Choosing this r
and taking x+ = rP + qy+, we obtain
d(X+; q, rP) ~ exp( -uC;u - C~' u) - 2rP /x+,
which yields the desired estimate, since by (11.4) and (3.6),
x+ = O(x exp( -Vlogx))
-rP «
x+
+ qy(1 + O(ljlogy)) =
~
exp( - y logx)
«
x(1
+ 0(1/ 10g2 x)),
exp( -u log z) ,
and by (3.2) and the definition 01 (x, y),
uC;u =
(1 +
0C:::) )UlOgU
= (1 + o (10g2 U)) 10
logu
= (1 + 0C:::)
)0
gy
1 (x,
log(logyjlogz)
logz
y) logy.
Remarks. Under the Generalized Riemann Hypothesis, (11.1) holds with the
error replaced by 0(PX e- I / 2 ) and, in the above proof, we may choose z =
(1/2)10gx leading to the conclusion of Theorem Al with 01(X, y) replaced by
o (x
3
,y
) = 10g(10gy / 10g2 x)
10g2 X
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J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
80
and the larger range q(logq)l+e:::; X:::; qexp((logq)1/2-e) in place of (2.2).
As remarked in §2, we can use a modification of the above argument to delete
the factor log2x from (2.12) and (2.13) of Theorem A3, at least for sums of the
form E q ...... Q max(a, q)=l' E q...... Q min(a, q)=l • Indeed, since we are able to assume
logy « log2 x log3 x, the estimate of Corollary 3.1, in conjunction with (2.14),
is sufficient for the proof. In this case we must choose y + above so that qy+
is independent of q and, for y this small, Corollary 3.1 allows us to do that.
Let y
= xl q
12. PROOF OF THEOREM A2
and u
= log y /log(log x log y)
(12.1 )
N(e) « u :::; log
so that
1/3
x.
Define v to be the positive real solution of
[( 2 +
(12.2)
1~~2VX) 10gXr = y
and let w = vllog2 x. We pick j = [v] - 1 or j = [v] so that j is odd when
we wish to prove (2.6) and even when we wish to prove (2.7). Clearly, if N(e)
has been chosen sufficiently large, then Ij - ul < eul2 .
Now take 1= y 1/'}/log x , h = 210gx, and z = (l + 2)logx, so that y =
(z - h)} . As
l/v
(
1 . 1v
v-
.
)
1= -IY Y /}- / = (2 + IOw)exp - ._J logy ,
JV
ogx
we have
(12.3)
2 + 10 10: 2 x :::; I « w exp(4Iw) «logl/2 x,
provided that N(e) is sufficiently large. Therefore,
(12.4)
/(1
+ 21l)}
:::;
/e
2jf / :::;
(logx)2/3(logx) 1/5 = (logX)13/15
as 2jll:::; tlog2x by the first inequality in (12.3). Let
= 1 + [clogxll0jlog2xlog(llog2x)]
as in Proposition 11.1 with e = 1/2. Then, by the last inequality in
k
with c
(12.3), and by (12.4),
k _ . + 1 > ~ log x _.
J
- 20 j(10g2 X)2 J
> ~ (logX)2/15 j (1 + ~)}
- 30 (log2 X)2
? 4jz) Iy
so that
( 12.5)
}
4j}
(z-h) =y? (k_j+l)z.
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I
OSCILLATION THEOREMS
81
Let n be the product of any k + 1 primes in (z - h, z] that do not divide
q. Note that, as Z5/8 ::; h ::; z, Huxley's theorem [Hu] gives n(z) - n(z - h) '"
hi log z and so there are at least v(q) + k + 1 primes in (z - h , z]. Choose P
as in Proposition 11.1, with e = 1/2.
Define the matrix L exactly as in the proof of Theorem AI, except with
y + replaced by y. By counting down each column, we see that the sum of the
elements of L is
RP {I + O(e -clogx/logP + e-c.jIOgXn
n9 rp(P)
L:
(n,P)=1
by Proposition 11.1. Now
c logx > clog x > 9 '10 (110 x) 10g2 x
10gP - klogz - J g
g2 log(llogx)
~ 2j log(/ Ilog2 x)
by (12.3), and cJlogx ~ 2j 10g(/1l0g2 x) . Therefore the sum of the elements
of L is
(12.6)
R(y+
rp~/p(y)){1 +0(1/(/1l0g2x)2Jn.
There are
R¢(q)
q
+ O(r(q))
= Rrp(q) {I
q
+ 0(1 I (/llog
2
X)2J n
row numbers r, with r '" Rand (q, rP) = 1; therefore we may choose some
such row (say row ro) that contains more than (respectively less than)
q (
P
){
I .2
2}
rp(q) Y + rp(p)rp(Y) 1 + 0(1 (j 110g2 x )
n
primes, in order to prove (2.6) (respectively (2.7)).
Let Xo = x± = roP + qy and a = a± = roP' so that (a, q) = 1 . Now, as in
the proof of Theorem AI, row ro contains exactly (j(xo; q, a) primes so that
(12.7)
(-1)
J-I.
~(xo,q,a)~(-I)
J-I
rp
P rp(Y)
(
1
)
(P)-+O
,2
2"
Y
(j 1l0g2 x) 1
By Theorem B2 and (12.5) we have that
(_I/-lrp(y)
y
~ !(~)l»
4 J
»
where c'
( 12.8)
(/1l0g2 x
= ce120, and so, by (12.7),
(-1
zl
_1
v7
(~k)J
JZ
;~g(llOg2 x)
r'
/-1 ~(xo ; q, a) ~ 1I (U 2110g2 X)(IH/3)u
~ 1/(U 3(IH/2)u 10g5 x)
by the second inequality in (12.3).
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J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
82
Finally, as log u = {I +
0 ( I)}
log(log Y /1og2 x) , we see that
3(1 + ej2)u log u :::; (1 + e)c52 (x, y) logy,
and so (2.6) and (2.7) both follow from (12.8).
Remarks. In general this is the best possible result, using this method, up to the
power of logx in the denominator. If logy:::; (1og2 X)2 then u :::; 10g2 x and
so the factor exp(4jw) in the upper bound in (12.3) becomes relevant. It does
not seem possible to remove this factor because of the I U in the denominator
of the right-hand side of (12.8). More precisely, in order to use Theorem B2,
we need I ~ 2 and so
>1
lil.J
o!; x >
_ y >_ (/+2/logix
k
_ ogi-I x.
Now in the proof above we need two values of j, one odd and one even, so if
y = logl x-I for some given integer J , we can take j = J and j = J - 1 only
if I ~ (1ogX)I/(/-I). But then we are guaranteed the factor I U ::::::: (1ogX)I+o(l)
in the denominator.
If we only need to get a good estimate on one side (i.e. for 1.:1(x; q, a)l)
when logy is small compared to (1og2 X)2 , then we do this with j = 1 +
[log y j log( 2 log x)] and I = 2. Select h, k , and z as in the proof of Theorem
A2, and then we can get the lower bound Ij(u2 log2 x)U(IH).
If we were to suppose that the error term in Proposition 11.1 could be taken to
be O(PX B- I / 2 ) (which would follow from the Generalized Riemann Hypothesis) then, in the proof of Theorem A2, we could take k = 1+ [log x j 1000 10g2 x] .
This would imply that, in the definition of c52 (x, y), we could replace the "3"
by a "2" and (2.6) and (2.7) would still hold.
As remarked in §2 we can also use Theorem B2 to prove a slightly weaker
version of Theorem A3 and its Corollary. By just considering those integers q
in [Q, 2Q] containing no prime factor in [2 logy , 410gy] and by taking 1= 1
in the proof of Theorem A2, one can, for example, deduce (2.15) in the range:
Q~xexp(-(I-e)B(1og2X)
2
/1og3 X) ,
where A = 3B + 5 .
13. PROOFS OF THEOREM A3
AND
PROPOSITION 2.1
We require the following simple result.
Lemma 13.1. For any nonzero integer a and all x
I:
(13.1)
m<x
~
2, we have
~~) = XCI (a) + O(r(a) log x) ,
(m,a)=1
( 13.2)
I:
m<x
1
~(m)
= c l (a) logx + c2 (a) + O(r(a)x -I logx) ,
(m,a)=1
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OSCILLATION THEOREMS
where
c, (a) =
and c2(a)
«
rp(a)
-a-
IT (1 +
pfa
83
1)
( _ 1)
PP
r(a).
Proof. It suffices to prove (13.1) since (13.2) then follows by partial summation.
Writing mjrp(m) = L.dlmIl2(d)jrp(d) and inverting the order of summation, we
have
~=
m<x rp(m)
(m,a)=,
2
L
L
11 ~)
1.
d'5.x rp() r'5.x/d
(d,a)='
(r,a)='
Since
"
1 = ~ rp(a)
~
d a
r'5.x/d
(r,a)='
+ O(r(a))
'
it follows that
L
m<x
(m,a)=,
m
rp(a)
--=x-rp(m)
a
L
d<x
(d,a)='
l(d)
d (d)
rp
+ O(r(a) logx).
The sum over d may be extended to a sum over all d prime to a. The error
in doing so is absorbed by the error O(r(a) logx). Once this is done, the main
term is just xc,(a) and we have (13.1).
Proof of Theorem A3. It is well known (cf. [Da, p. 124]) that, with a suitable
constant c, 0 < c < 1 , the estimate
8(x; r, a) = xjrp(r)
+ O(xexp(-2cVlogx))
holds uniformly for all r :s exp(2vlogx) and all integers a with (a, r) = 1,
except possibly for those moduli r that are mUltiples of a certain exceptional
modulus r, > 1 . It follows that if a is a nonzero integer satisfying (a, r,) > 1
(if r, exists), then uniformly for R, :s R2 :s exp(cvlog x ) , x'/2 :s x, :s x 2 :s
x, we have
(13.3)
"~ {8(x 2 ; r, a) - 8(x,; r, an = ~
"
x ~(r)
-x , + O(x exp( -c, V ~
logx))
<r'5.R2
(r,a)='
Rj
R j <r'5.R2
(r,a)='
for any c' < 2. Now let xexp(-cvlogx):S Q,
:s xj2
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and 0 <
lal <
Q,.
J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
84
Then
L
8(x;q,a)=
=
L L
q,,-,Ql a<p$.x
(q, a)=1 p-a=qr
10gp+O(Qlloglal)
L
{8(a+2rQ 1 ;r,a)-8(a+rQ 1 ;r,an
r$.x/2Q1
(r,a)=1
L
{8(x; r, a) - 8(a + rQ 1; r, an
r"-'x/2Q1
(r,a)=1
+ O(Qllog lal) + O(QI log2(2x/QI))'
where the first error term absorbs the contribution of the primes ~ a to
8(x; q, a) and thus, in view of the bound lal < Q 1 < q, may be dropped
unless a itself is a prime. The latter error term accounts for the replacement
of (x - a) by x in the ranges of summation. By (13.3) this implies
+
'"'
L..J 8(x; q, a) = '"'
L..J rQ 1 _ '"'
L..J rQ 1
q,,-,Ql
r:5.x/2Q1 qJ(r) r"-'x/2Q1 qJ(r)
(q,a)=1
(r,a)=1
(r,a)=1
L
qJ~r) + O(QIlog(lalx/QI))'
r"-'x/2Q 1
(r,a)=1
> 1 in the case that the exceptional modulus r l exists.
+
provided that (a, r l )
Setting
Sa(t)
=
we can write the right-hand side as
L
r<t
r/qJ(r) ,
(r,a)=1
QISa(2~J -QI(Sa(~J -Sa(2~J)
1
+ x L qJ(r) + O(QIlog(lalx/QI))'
r"-'x/2Q 1
(r,a)=1
which, by partial summation, reduces to
x / Q1
dt
x
Sa(t)2" + O(Ql 10g(l a lx/QI))'
x/2Q 1
t
Since
l
L
q,,-,Ql
(q,a)=1
8(X;q,a)=x{
L
q,,-,Ql
(q,a)=1
qJ!q)+
= x{ c1(a) log 2 +
L
q,,-,Ql
(q,a)=1
L
q,,-,Ql
(q,a)=1
qJ(1q )d(X;q,a)}
qJ!q)d(X; q, a)}
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+ O(XI/2)
85
OSCILLATION THEOREMS
by Lemma 13.1 and the bound r(a)« lal'/3 :::; Q:/3 :::; X'/3, it follows that
(13.4)
L
q~QI
_l_Ll(x; q, a) =
rp(q)
jX/QI (Sa(t)
x/2Q 1
_ tc, (a)) d;
t
+ o(Q,IOg(la1x/ Q ,)).
x
(q,a)='
Now, let x and Q be given subject to (2.11) and let y = x/Q. We give the
proof of (2.13), the proof of (2.12) being almost identical. Set z = (1/2) logx .
In case there is an exceptional modulus r, :::; exp( y'logx) ,let p be a prime divisor of r, ' otherwise let p = 2. By Theorem B3 there exist z_ in the range (3.4)
and y_ in the range (3.6) so that ;:p(y_, z_) satisfies (3.7). Let Q_ = x/y_
and a_ = p I1 p '<z_ p'. By (3.6) Q_ satisfies Q :::; Q_ :::; Q(l - 1/ logy)-' ,
and thus, in view of (2.11), lies in the range xexp(-cy'logx) :::; Q_ :::; x/2.
Moreover, we have a_ f:. 0, Q_ > laJ, and, if r, exists, (a_, r,) 2: (a_, p) =
p > 1 . Hence the above argument applies with Q, = Q_, a = a_ , and (13.4)
holds. Now it is easily seen that, with the above choice of a_ ,
j
X/Q_
xj2Q_
dt
(Sa_ (t) - tc, (a_)) 2 = c, (a_);:P (y _ , z_).
t
Thus, (2.13) follows from (13.4), (3.7), and the estimate
II (1 _pi..!..) » _1_
» _1_.
log z _
log2
c (a )>> rp(a_)>>
,a_
,
X
p <z_
Proof of Proposition 2.1. In the case Q:::; X 2/ 3 , the result follows at once from
Theorem 9 of [BFI] (for fixed a) or from Fouvry [Fo]; in fact, their results
apply in a much larger range.
In the case Q > x 2/ 3 , we follow the above argument except that instead
of using the estimate (13.3) we bound 8(x; r, a) on average by means of the
Bombieri-Vinogradov theorem. In this way, we obtain (13.4) with Q, = Q and
an additional error term OA((logx)-A) on the right-hand side. Since by (13.1),
r/Q(Sa(t) - tc,(a))d; =
}Xj2Q
t
o(
r/ Qr(a)IOgt d;) = o(r(a) QIOg(X/Q)) ,
}X/2Q
t
x
this proves the desired estimate (2.16).
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[CG)
(J. Friedlander, A. Granville) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO,
TORONTO, ONTARIO M5S lAl, CANADA
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS, URBANA, ILLINOIS 61801
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF GEORGIA, ATHENS, GEORGIA 30602
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