Math. Proc. Camb. Phil. Soc. (1991), 110, 337
Printed in Great Britain
337
Effective mean value estimates for complex multiplicative functions
BY R. R. HALL
Department of Mathematics, York University, Heslington, York Y01 5DD
AND G. TENENBAUM
Departement de Mathematiques, Universite de Nancy I, BP 239,
54506 Vandoeuvre Cedex, France
(Received 7 December 1990; revised 4 February 1991)
1. Introduction
Quantitative estimates for finite mean values
x-1 s g(n)
(l)
n^x
of multiplicative functions are highly applicable tools in analytic and probabilistic
number theory. Extending a result of Hall [4], Halberstam and Richert[3] proved
a useful inequality valid for real, non-negative g satisfying for instance a Wirsing
type condition, viz 0 ^ g(p") < A^JT1 (v = 1,2,...) for all primes p, with constants
Aj ^ 0, 0 ^ A2 < 2. Their upper bound is sharp to within a factor (l+o(l)), but
even a weaker and easier to prove estimate, such as
x-1 S 9(n) < exp{- 2 A z M l
n^x
I
p ^x
P
(2)
J
(where the implied constants depend on Ax and A2), may become a surprisingly strong
device. For instance, setting g(p) = l ± e , where e is an arbitrarily small positive
number, provides immediately a proof of the famous Hardy-Ramanujan theorem on
the normal order of the number of prime factors of an integer. This example, and
many others, are discussed in detail in our book [5] where we make extensive use of
(2) for various problems connected with the structure of the set of divisors of a
normal number.
Hildebrand [6,7] refined the Halberstam—Richert inequality and obtained
corresponding sharp lower bounds. In another direction, an inequality of
Tenenbaum([10], theorem III-4-7) states that, when g: Z + -> [— 1,1], the estimate (2)
remains valid provided thep-sum is multiplied by g. The proof rests on Montgomery's
effective version of Halasz' mean value theorem (see Lemma 1 below) and the
possibility of deducing a result of this sort had been indicated by Montgomery [9]. A
slightly weaker version appears in Elliott[1], who uses a different method.
It is natural to ask whether there is a similar inequality under the weaker
hypothesis \g\ ^ 1, perhaps with a smaller constant and of course 3ieg{p) appearing
338
R. R. H A L L A N D G. T E N B N B A U M
in the exponential. The example g(n) = nl shows that this is not the case, for the lefthand side of (2) is then Q(l) whereas
s
g2
P«x
P
(Here, and in the sequel, we let logfc denote the k-th fold iterated logarithm.)
Nevertheless, we do have a result of this type if we impose an extra condition on g.
At first sight this might appear artificial - in fact it is quite practical.
For 0 ^ 8 ^ 1, 0 ^ (j> < n, we let S(8, <fi) be the set of complex numbers z with
(gro (e-^z))2 ^ 82{l - (We (e^z)) 2 },
(3)
(
and we denote by S{8,<1>) the class of all multiplicative functions g such that
\g(n)\ ^ 1 for all n and g(p) e $(8, $) for all primes p. We shall also make use of the
linear mapping W defined by
W(z) = ef*{<Re (e~^z) + i8%m (e'^z)}.
(4)
We have the following result.
THEOREM.
For 0 < d ^ 1, 0 < <j> < n, the integral equation
W{ea-K)\d6^1-K
(5)
has a unique solution K = K(8,<j)) ^ 0, which is, for fixed (f>, a decreasing function of 8
such that K > 0 if and only if 0 ^ 8 < 1. We have uniformly for x ^ 1 and ge'S(8,(j))
S g(n)<xexJ-K(S,<t>) £ l~*e9{p)\
(6)
Moreover the constant K(8, <j>) is sharp: given (8, <j>) e [0,1] x [0, n[ and x ^ 3, there is a
ge&{8,<t>) such that
P
(ii)
| S
-K(8,<t>)
£
g(n)\>xexJ-K(8,<t>)
nix
I
vix
^ ) \
P
)
We prove the statements on K(8,<f>), as well as some extra remarks, in Section 4
which also contains a table of K(8,0) computed by M. G. J. van der Burg.
Condition (i) is trivially realized unless 0 = 0 or 8=1- indeed
We g(p) < y/ (cos2 <fi + 82 sin2 <f>)
for ge(S{8,<f>). When 8 = 0, <f> = \n, we have W(eie—K) = isind for every K. Hence
K(0, \n) = 1 — 2/77, and the theorem implies that
max
I 2J g(n)\ —
The case 8 = <f> = 0 of the theorem corresponds to real functions and (6)
Complex multiplicative functions
339
provides then the sharp form of Tenenbaum's inequality. It is easily checked that
K(0,0) = 0-32867... = — cos^ 0 , where <j>0 is the unique root in (0,n) of the equation
s i n 4> — <j) c o s <f> = \ n .
Quantitative estimates of the form (6) already appear in Halasz[2], with a
different condition on the values g(p). He notes in particular that, for completely
multiplicative functions g with values + 1 , his main theorem yields a bound of the
type (6) with a constant which can be taken as large as 0-07 but not 0-37.
Conditions (i) and (ii) guarantee that the right-hand side of (6) cannot be
multiplied by a function having lower limit zero as Zip^x(l — 9ieg{j)))p~1 tends to
infinity. Thus K(S, <j>) is sharp in the strongest possible sense. We imagine that our
lower bound method is partly close to the lower bound derivation of an unpublished
result of Montgomery and Vaughan
/i(d)\ >zx (log x^11"^1.
max | £
n?l
(7)
d^x,d\n
For a fixed function g, the estimate (6) may lead to an unsharp bound. Consider
for instance g(ri) — T(n,d)/r(n) where r(n, 6) ••= Zid\ndlB. This last function occurs in
the theory of Hooley's A-function and in the proof of the Maier-Tenenbaum theorem
[8] - both applications being developed in [5]. We have g(p) = 1(1 +pie), so g(p) lies
in the ellipse ^(1/^/2,0) and (6) yields
g(n)
(\d\logxf
uniformly for 1/logx ^ \6\ < 1 (say), with K = K(\/y/2,0) = 0-135,..., whereas it
can be shown directly by contour integration that the correct exponent is K = \. This
example raises the problem of finding other conditions on g(p) (i.e. defining a different
class of functions), for which there is still a sharp general inequality but with less
possible fluctuations in the result. •
2. Proof of the upper bound result
In this section, we prove (6). We require the following lemmas, the first of which
is the inequality of Montgomery [9] referred to in the Introduction.
LEMMA
1. Let ge&(l,0). Set
G{x)-.= 2 g(n), F(s)-=2g(n)n-°.
n ^ x
(8)
n—1
For a. > 0, let H(a) be defined by
H(a)2--= £ (k' + l)-1 max \F(l+a + ir)\2.
Then
G(x)^-^—\
•ogzJ
H(a) — .
(9)
(10)
a
Montgomery restricted his attention to completely multiplicative g. The more
general case involves only technical changes. For details, see [10] chapter III-4.
340
R . R . H A L L AND G. T E N B N B A U M
LEMMA 2. Let fbea periodic function of bounded variation over the period [0, 2n] and
having mean value
-If2"
f-=—
f(v)dv.
J
JK
2TTJ
'
Then for real T, W such that T 4= 0, 0 < W ^ Z and every positive c, we have
(U)
where
M(f) == sup \f{v)\,
V(f) ••= P" \df(v)\.
v
Jo
-
This is lemma 30 l of [5]. We state the following immediate consequence of
Chebyshev's inequality n(t) <^ t/logt for convenience of reference.
LEMMA
3. Let 0 < a ^ 1. Then
£
p<exp(l/a)
p-1'* < I-
1^1+ £
P
(12)
p> exp(l/a)
Recall the definition of W in (5) and set
T{z) ••= e** {«e (e-^z) + iS-^m (e^z)},
(13)
where, by convention, the term involving £~x is to be deleted if d = 0. The functions
T and W may be regarded as linear mappings on C considered as a real vector space.
L E M M A 4. For 8 = 0, the restrictions of T and W to g(8, (p) both coincide with the
and W-.S1 (1,0)^£(8,0)
are
identity. For 8 + 0, the restrictions T:g(8,<j))^S(l,0)
reciprocal isomorphisms. Moreover, when 8 4= 0, we have
)W(z2))
(ZleC,z2eC).
(14)
Proof. The case <j> = 0 is trivial and we suppose henceforth that 8 > 0. We could of
course check the various statements by direct explicit computation, but it will be
more satisfactory to make an algebraic reasoning. Let R(</>) denote the matrix of the
rotation of angle <p and D(t) the matrix of the dilation (£, ?/)i—>(£, tv). ThenD(i) induces
a one-to-one mapping from S(1,0) to S{8, 0) and D{t)~l = D(t~l). Moreover, T and W
are respectively associated with R(<j>)D(8~1)R( — <j>) and R((j>)D(8)R( — (j>). Since the
unit disc S(i,0) is invariant under R(<j>) and-R(^)"1 = R( — <f>), we immediately obtain
the first assertion. Next, let Zx, Z2 be the column matrices associated to the complex
numbers zltz2. Using a dash to indicate transposition, we have
We(T(Zl)W(z2))
=
= Z'xR(-<t>)' D(8-lYR(<fiYR(</>)D(8)R(-<t>)Z2
= Z'1Z2 = me(z1z~2),
since R( + <f>)' = R( + <f>) and D(8~1) is symmetric. This establishes Lemma 4.
Complex multiplicative functions
341
We may now embark on the proof. We begin by establishing that, uniformly for
(<S,0)e[O,l]x[O,7T[,O<as$ 1, re05, we have
i | H V T - K ) \ ^ (1 -K) log- + 0(loga(M + 3)).
2
(15)
a
IXexpd/a)?'
We set
f(v):=\W(eiv-K)\.
When |T| ^ a, we have
= \W(l-K) + 0(Tlogp)\ ^ l-K+0(Tlogp),
since W(l-K)
immediately
(16)
= (l-K) W(l) and 0 ^K ^ 1, |W(1)| ^ 1. Hence if \T\ ^ a, we have
S
-/(Tlogp)<(l-Jr)log-+0(1).
(17)
a
p « exp (I/a) P
Next, let a < |T| ^ 1. We put w==exp{l/|T|}. By (16), we have
2 -f(Tlogp)^(l-K)log2w + 0(l).
(18)
Now, we put 2 ~ exp (I/a) > w and apply Lemma 2. We have /"= 1— K by (5),
1. V(f) < 1. and (11) gives
S
- / ( T l o g p ) ^ ( l - i r ) ( l o g - - l o g 8 t i > ) + 0(l) >
(19)
so that (18) and (19) imply (17) in this case.
Suppose now that \T\ > 1 but log2(|T| + 3) ^ I/a. We select c = 1 in Lemma 2,
w;:=exp{log2(|T| + 3)}, z-= exp (I/a). This gives (19) and we make the trivial estimate
•^ log2w; for the sum in (18). So we obtain (17) with 0(log2(|T| + 3)) in place of 0(1).
Finally, if log2(|r| + 3) > I/a, we estimate the whole sum trivially. Hence (15) holds
in all cases.
Let us now consider (d, 0)e[0,1] x [0, n[, and ge^(8,(f>). For ae[0,1], we define
A = A(a) by
p«exp(l/a)
P
p< exp (I/a) P
so that Ae[0,2] since \g(p)\ ^ 1. We claim that (20) implies
(21)
p « exp (I/a)
uniformly for relR. For this we need a device which forces us to consider elliptical
sets in the theorem rather than some other family of ovals. For primes only, define
the function h(p) by
h(p) = T(g(p)).
(22)
By Lemma 4, we have \h(j))\ ^ 1. Also (14) with z1 = g(p), z2 = 1 gives
342
R.
R.
H A L L AND G.
TENENBATJM
and (20) may be written as
1
1
— A(p) W(i) = (1 — A)log—\-0(l).
2
(20'
Now we use (14) again to obtain
From this and (20') we deduce that
9^
9ip)p~1~ir = We
2
p«exp(l/a)
-h(p)W(piT—K) + (l —A) K log —
2
a
p < exp (1/ot) #
(23)
Using the inequality yie{h(p) W(ph-K)} ^ \W(pir-K)\ and applying (15) yields
immediately (21).
Now consider the function F defined by (8). We have
J T (l+a + i T ) ^ e x p ^ e 2 \zL- \
I
p
(24)
p1+a+n)
and, by Lemma 3, we may restrict the p-sum to the range p ^ exp (I/a) and strike
out the pa. We recall (20) and deduce from (21) that
F(l + <x + iT) -^ aKA~1logB(|T| + 3)
(25)
uniformly for 0 < a ^ 1, re U, whereB is an absolute constant. Hence Montgomery's
function H(a) defined by (9) satisfies
TJ i
\2 ^
2
Iocr
^!&l + 4^
& M l "^ /
2u
r^—j
2KA—2 V
c
rt(a) <5 a
/ctn\
(^o)
H(a) « a^" 1 .
and so
(27)
Now let A e [0,2] be defined by the equation
= A
V <x
i3
2 - = Alo g2 x + 0(l).
(28)
P Sit?
Then we have for 1/logx ^ a.
p«exp(l/a)
?*
p«s
P
exp (I/a)
()
(29)
a
We deduce from this and (20) that
(30)
Whence we obtain from (27)
B(oc) < a.2*-1 (logx)(2-A)K.
(31)
Complex multiplicative functions
343
Inserting this in (10) and using the fact that K ^ K(0, TT/2) = 1 — 2/TT < | , to be
proved in Section 4 - equation (60) - , we obtain
G(x) <-^—(logx)1-2K+^-A)K
= x(loga;)-AK.
Recalling the definition (28) of A, we obtain (6).
3. Completion of the proof of the theorem
We now show that the constant K = K(S, (p) is sharp for all (S, <fi) 6 [0,1] x [0, n[ by
constructing for every x > 1 a function ge^(S,(p) which satisfies conditions (i) and (ii)
of the theorem. We first introduce some further notation
P(n)--=maix{p:p\n} (n>l),
P(l)==l,
V
L(x)-= 2 g(n)logn {g
n^x
We need two lemmas, the first of which arises as an intermediate step in
Montgomery's proof of Lemma 1 - see [9] or [10], p. 380.
LEMMA
5. Let ge^(l,0).
Then we have
1/logj/
a
6. Let w ^ 2 awd ge@(l,O) be supported on squarefree numbers and such that
g(p) = 0 for p > w. Then we have
LEMMA
G(t)\ogt-L(t)<texp 1 - ' ° g l \
{
2 log w)
> 9 in > 9\
IW\
£Z &j W ^
\OO)
r m^waL. w
Jw
where
l
V(a0log2/)
Lif
(34)
a 0 == max I;
,
1.
\\ogw logy/
Proof. The left-hand side of (33) is plainly
<
2
log (t/n).
nit
Pin) s; w
The required estimate hence follows, by partial summation, from the estimate
proved in [10], theorem 111-51.
344
R.
R.
H A L L AND G.
TENENBAUM
The proof of (34) consists in a reappraisal of part of the proof of Lemma 1, where
we take into account the extra hypothesis that g(n) = 0 whenever P(n) > w. In view
of (33) and the estimate H(a.) > 1 (see [9] or [10], p. 375) it is enough to show that
\L(t)\
/log?/
it < H(cc0)
t2
(35)
Now we write Cauchy's inequality
(36)
and apply the Plancherel identity with <x— I/log?/
in (36) does not exceed e2 times the left-hand side
Since the integral involving
of (37), we obtain
.
(37)
dr.
l+a + ir
F'
max
—
dr.
(38)
By a lemma of Montgomery [9] the last T- integral is
logp
l+a+jr
(39)
dr.
But one can easily prove by standard contour integration (see e.g. [10], pp. 418-20)
that for s = l+ct + ir, \T\ <§ 1,
w
1-8
5-1
P
min
1
8-l\
,
This shows t h a t the integral in (39) is
F(l+a + ir)
logw;
l
1
. We also notice that
•4 exp
Collecting the above estimates and inserting in (38), we obtain
i:
I n view of (36), this implies the required bound in (35).
We are now in a position to embark on the final p a r t of the argument. We may
restrict ourselves to the case S < 1 since K(l,rf>) = 0 and, as we remarked in the
introduction, g{n) = nl provides the desired counter-example for 8=1. In the
remainder of this section we hence let (8,<j>) be fixed in [0, l [ x [O,TT[. The variable
Complex multiplicative functions
345
x being given, we introduce a parameter iv = w(x) to be chosen explicitly later. Let
g1 be supported on squarefree numbers and gt(p) = 0 for p > w. We put
(40)
l i ^^
where, by convention, the right-hand side is to be interpreted as 0 when the denominator vanishes - this case being possible only when # = 0. Since g1(p)eW(& (1,0)),
we have g1e^(S,<f>) by Lemma 4.
We first show that
S1{w)= 2
^ - ^ + oo (w^+co).
J>«HJ
(41)
P
This is trivially satisfied when <fi #= 0, since then (^egfjtp))2 ^ cos2 <j> + S2 sin2 <j> < 1
for all p . For <f> = 0, let f^d) be the periodic function of d defined by ft(6) = 1 if
cos (9 < if, Z^/9) = 0 otherwise. Then
for all £> and Lemma 2 implies (41) since
/x = 1
ArccosX ^ \.
77
Next, we prove that, with an obvious notation,
>
\MAdt>e-KSlmiogw.
(42)
For this, we notice on the one hand that
f+OO
+ ») f
;orM#,
(43)
n-l
and on the other hand that
(44)
n-l
But, by Lemma 4, we have for all p
p).
(45)
Hence it follows from Lemma 2 and (5) that
P
p ^W
= log2w-KS1(w) + 0(l).
Together with (43) and (44), this yields (42).
P
(46)
346
R.
R.
H A L L AND G.
TENENBATJM
We now show that (42) remains valid when the <-range in the integral is restricted
to wc<> ^ t ^ u>c> where c0 and cx are suitable absolute constants, that is
(47)
We need to show that the contributions of the ranges 1 < t < wc° and t > w°l are
relatively small compared to the value of the whole integral in (42). For the first
range, we apply (32) and (31). We obtain, with y = wc«, c0 < 1,
Since K ^ 1 — 2/TT <\ — see Section 4 - we obtain that, for suitable c0,
aiau,.
m
For the second range, we apply (34) to qx with y = yk = wCl2 , k = 1,2,..., and sum
the resulting estimates. This gives, with a0 = I/log w,
fc-i
y/cx
where we used (31) in the last stage. Thus, for suitable cx, we obtain in view of (42)
™
l
*
(49,
The estimate (47) now follows from (42), (48) and (49).
The constants c0 and cx being fixed, with c0 < 1 < cx, let $£{w, y) denote, for
positive y, the set of all t such that
wc" ^ t ^ vf\
\Gx{t)\ > yte'KS^w).
(50)
Since, by (6), we have
\G1(t)\^c2te~KS^
for all t in [wc<>, wCl] and suitable absolute c2, we have
f M * < ye-«.« f
* + c,e-«.<-> f
*.
(51)
But the first ^-integral on the right does not exceed ct log w, and we may hence infer
from (51) and (47) that there exists an absolute constant y > 0 such that
dt
— >\ogw
(52)
Complex multiplicative functions
347
—
(53)
and hence
J*(w.
> 1.
We now reach the last part of the proof. We fix x large and put w = x1/2Cl. Since
^t) — G^s)] ^ l+s — t for 1 ^ t ^ s, we plainly have
tesf(w,y)
implies
MU+j——)
-
Thus there is a finite sequence tltt2, ...,tr with ti+1/t} ^ (1 + I / l o g 2 #) for all j , such
that
sf(w,y) S U [ ^ ' ^ ( 1 + i ^ ) ] - ^(w>h>)-
(54)
In particular it follows from (53) that
log8* £ J t ,
However, by the prime number theorem, we have for every j = 1,2, ...,r
(56)
It therefore follows from (54), (55) and (56) that
We split the above sum into four parts corresponding to the different possibilities
of signs for We^G^x/p)}
and <5{e{ei't'G1(x/pj\ — !$m{ei't'Gl(x/p)}. One of these is
at least \ of the whole sum and we admit for instance that it corresponds to
SRe {e^G^x/p)} ^ max (0, ^mle^G^x/p)}). Thus we obtain
S<w->1
P
(58)
IT
where the asterisk denotes that summation is restricted to those p such that
x
,
.
w < — ^ w x = \/x
V
p. Zg-KS^x)
We now define a multiplicative function g2, supported on squarefree numbers, by
92(P)=9i(P), for
p^w,
g2(p) = e^,
for p counted in (58),
g2(p) = 0,
otherwise.
348
R . R . H A L L AND G. T B N E N B A U M
Plainly, gr 2 e^(5, <f>). Since condition (*) implies p > \/x, we have
n ^ a:
n <x
p
n < x/p
From condition (*) and (58) it follows that
Hence either grx or g2 fulfils conditions (i) and (ii) of the theorem, and the proof is
thereby completed.
4. On the function K(8, (/>)
For Os$0=$7r, 0 < <$ < 1, 0 < .K" < 1 define
G(A) = G(K,S,4>):=K+^-
?n\W{ei0-K)\d6.
(59)
For ^ = 1, we have W(eie—K) = ete—K and iC = 0 is the only root of the equation
G(K) = 1. We hence assume from this point on that S< 1. We have with this
hypothesis
hence (?(0) < 1 < 6?(1). Moreover, the triangle inequality implies that, as a function
of K, \W(eie-K)\ = \W(eie)-KW(i)\ satisfies a Lipchitz condition with modulus
| W(l)|, so dG/dK > 0 unless perhaps when <f> = 0. In this latter case, the Lipchitz
condition is strict unless 6 = 0(mod7r), whence again dG/dK > 0. We may therefore
define K(S, <j>) to be the unique solution of the equation
We also observe that, for any fixed z,
\W(z)\ = V(^e(ze^)2 +
in an increasing function of d, hence dG/dd ^ 0 and K(S, (f>) is a decreasing function
of 8.
Our main purpose in this section is to show that for all 8, <j> we have
K(8,4>) ^ K(0, \n) = 1 - 2/TT
and
2
2
\{i-8 )^K(8,4>)^\(\-8 ).
(60)
(61)
We begin with (60). Since K(8,<p) ^K(0,</>), we only have to prove that, for any
fixed K, G(K,0, <j>) is minimal when <f> = n/2. But
(*
J
^~
(\cosd-Kcoa$\dd
in Jo
Complex multiplicative functions
349
with /?:= Arccos(Xcos^). The function of /? between curly brackets attains its
minimum value 1 when ft = \TT, which corresponds to </> = \n, and this is all we need.
Next, we show (61). To this end, define, for r ^ 0,
M(r) = J - \\W(ei0-K)\rdd.
^n Jo
(62)
M(2) = |(1 + 82)+K2(cos2<j> + 82sin2^>) ^\(l + 82)+K2.
(63)
We have
= 1 -K and by Cauchy's inequality,M(2) ^
(1-K)2.
It follows from (63) that
which is equivalent to the lower bound in (61). To obtain the upper bound, it will be
sufficient to show that
G{\(\-S%8,<j>)^i
(64)
M(1) ^ \{ 1+ 82).
or equivalently
(65)
By Holder's inequality, M(2) < M(l)2'3M(4)113 and so (65) will follow from
M(2)3^\(1
+ 82)2M(4:).
(66)
t--= (l-82)2sin2<f>.
Let us set
(67)
From (63), on substituting K = \{i — 82), we have
(68)
To compute il/(4), define
F(0) = \W(eie-K)\2
= M(2) - 2K (cos <f> cos (6 - <f>)-82 sin <f> sin (d-</>)) +±(1-82) cos 2(d-<p).
(69)
We have, from (62) and (69),
P)2,
(70)
2
and we substitute K = \(l — 8 ) and simplify, employing (67), to obtain
M(4) = M(2)2 + l(\-82)2-\(i-8i)t.
(71)
Therefore (66) is equivalent to
or
M{2)2(2-t)^{i-82)2{%(\-82)-\(Y-82)t}.
(72)
We substitute the right-hand side of (68) for i f (2) and arrange the desired inequality
as a cubic in t; we require that (after multiplication through by 8),
{(3 + S4)2 - 5(1 + 82) (1 -
8*)}-{¥-1882-IS4l(l-82)H3^0.
(73)
350
R. R. HALL AND G. TENENBAUM
Table 1
K(S,0)
0
005
01
0-15
0-2
0-25
0-3
0-35
0-4
0-45
0-5
0-55
0-6
0-65
0-7
0-75
0-8
0-85
0-9
0-95
1
0-32867...
0-326...
0-320...
0-312...
0-302...
0-291...
0-278...
0-264...
0-249...
0-232...
0-215...
0197...
0-178...
0-158...
0138...
0116...
0094...
0072...
0-048...
0-024...
0
(l-S2)/K(S,0)
3042...
3057...
3088...
3126...
3170...
3-218...
3-267...
3-318...
3-370...
3-423...
3-477...
3-530...
3-583...
3-637...
3-690...
3-742...
3-795...
3-847...
3-898...
3-950...
4
Since t ^ 1 the sum of the last two terms on the left is at least |(1 - 82) (l-82
and so it will be sufficient to show that
+ 28*) t2
^0
(74)
and we notice that when 82 ^ 1/3 this is trivially true because all three coefficients
are positive, and 0 ^ t ^ 1. Denoting the above quadratic by at2 + bt + c, we have
fc2 _ 4oc = - 55 - 40082 -10084 + 13048* + 10348s + 384810 + 112812 - 8814 + 816.
(75)
This is negative when 82 < 1/3. and the quadratic is positive for all t. It follows that
(64), and hence the right-hand inequality in (61), hold.
We append the table of K(8,0) (Table 1) kindly provided by M. G. J. van der Burg.
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