ON THE PAIR CORRELATION OF ZEROS OF THE
RIEMANN ZETA-FUNCTION
È ZLU
È K and C. SNYDER
D. A. GOLDSTON, S. M. GONEK, A. E. O
[Received 16 October 1997; revised 4 January 1999]
1. Introduction
In 1972 Montgomery [20, 21] introduced a new method for studying the zeros of
the Riemann zeta-function. One of his main accomplishments was to determine
partially the pair correlation of zeros, and to apply his results to obtain new
information on multiplicity of zeros and gaps between zeros. Perhaps more
importantly, he conjectured on number-theoretic grounds an asymptotic formula
for the pair correlation of zeros and found that the form of this correlation exactly
agreed with the Gaussian Unitary Ensemble (GUE) model for random Hermitian
matrices which had been studied earlier by physicists. He was therefore able to
formulate a general n-correlation conjecture for zeros. During the 1980s Odlyzko
[23, 24] performed extensive numerical calculations of the correlations for zeros
in ranges up to the 1020 th zero and found excellent agreement with the GUE
model. More recently Hejhal [17] was able to prove the same partial result for
triple correlation as Montgomery proved for pair correlation, and Rudnick and
Sarnak [25] have done the same for n-correlation. Rudnick and Sarnak extended
their results to a large class of L-functions, and also showed that the Riemann
Hypothesis (RH) is not needed for smoother forms of the asymptotic result. Very
recently Bogomolny and Keating [1, 2] have used a prime-twin type conjecture to
derive heuristically the n-correlation conjecture beyond the range where the results
of Rudnick and Sarnak apply. The conclusion of all this work is to ®rmly
establish (but not prove) the GUE distribution for zeros of many zeta-functions.
There is a dual relationship between zeros of the Riemann zeta-function and
prime numbers. Following Montgomery's work, it was realized that information
on pair correlation of zeros could be used to obtain information on primes. This
connection was developed by Gallagher and Mueller [10] and Heath-Brown [15].
Later Goldston and Montgomery [14] found an equivalence under the Riemann
Hypothesis between the pair correlation of zeros and the variance for the number
of primes in short intervals. This equivalence arises out of the explicit formula via
a Parseval relation, together with a Tauberian theorem. From this work one sees
that to obtain new information on primes from zeros will require some new
insight on the zeros and, while the connections to statistical physics mentioned
above are a possible source of this insight, so far no progress has been made on
this fundamental problem.
The motivation for this paper is the observation that there are additional tools
Research of the ®rst two authors was supported in part by NSF Grants.
1991 Mathematics Subject Classi®cation: primary 11M26; secondary 11P32.
Proc. London Math. Soc. (3) 80 (2000) 31±49. q London Mathematical Society 2000.
32
d. a. goldston et al.
available for the study of primes that do not arise via explicit formulas. In
particular, partial results in the direction of the twin prime and prime tuple
conjectures can be obtained by the circle method, sieve methods, and information
on primes in arithmetic progressions. Furthermore, primes can be studied by
means of combinatorial identities of Vaughan, Linnik, and Heath-Brown. One can
then exploit the equivalence between primes and pair correlation found in [14] to
prove new results about the zeros. Following this line here, we obtain a non-trivial
lower bound for Montgomery's function (which is related to the pair correlation of
zeros) in a range beyond that treated by Montgomery. The main tools we use are
an approximation to the usual prime indicator function that arises from the circle
method and a new mean value theorem for long Dirichlet polynomials and tails of
Dirichlet series [13]. Although we need to assume the Generalized Riemann
Hypothesis (GRH) for Dirichlet L-functions to control the error term in the prime
number theorem for arithmetic progressions, the result we obtain is, except for a
constant factor and a restricted range of applicability, the same as would be
obtained by applying a twin prime type conjecture. The signi®cance of this kind
of result is that we are able to go beyond the `easy' range with no (explicit)
number-theoretic assumption and only an assumption on the horizontal distribution
of zeros. Our result also allows us to improve a little on Montgomery's results on
the multiplicity of zeros and on small gaps between zeros (now assuming GRH
instead of RH).
The authors take this opportunity to thank Professor D. R. Heath-Brown for a
suggestion that led to a substantial improvement in the ®nal result.
2. Pair correlation of zeros
To study the distribution of pairs of zeros of the zeta function, Montgomery
introduced the function
ÿ1 X
0
T
log T
T i a…g ÿ g † w…g ÿ g 0 †;
2:1†
F…a† ˆ FT …a† ˆ
2p
0 < g; g 0 < T
where a is real and T > 2, g and g 0 denote the imaginary parts of zeros of the
Riemann zeta-function, and w…u† ˆ 4=…4 ‡ u 2 †. The normalization factor in front
of the sum arises from the Riemann±von Mangoldt formula
X
T
1,
log T as T ! 1:
2:2†
N…T † ˆ
2p
0<g<T
Let r…u† 2 L1 , and de®ne the Fourier transform by
Z 1
br …a† ˆ
r…u†e…au† du; e…u† ˆ e2p i u :
ÿ1
1
If br …a† is also in L , we have almost everywhere the inversion formula
Z 1
br …a†e…ÿua† da:
r…u† ˆ
ÿ1
On multiplying equation (2.1) by br …a† 2 L1 and integrating, we obtain
Z 1
X
T
0 log T
br …a†F…a† da:
r …g ÿ g †
w…g ÿ g 0 † ˆ
log T
2p
2p
ÿ1
0 < g; g 0 < T
2:3†
pair correlation of zeros
33
We can thus evaluate a large class of double sums over differences of zeros
provided we have an asymptotic formula for F…a†. The weight w…g ÿ g 0 † is not
signi®cant in (2.3) and can usually be removed. Our knowledge of the function
F…a† is limited at present. It is easy to see that F…a† is even and, by equation
(4.8), that F…a† is non-negative. That is,
F…a† ˆ F…ÿa†;
F…a† > 0:
2:4†
Montgomery proved, assuming RH, that uniformly for jaj < 1,
F…a† ˆ T ÿ2ja j log T…1 ‡ o…1†† ‡ jaj ‡ o…1†;
as T ! 1:
2:5†
This result allows us to evaluate the sums on the left in (2.3) over the class of L1 functions with Fourier transforms having support in ‰ÿ1; 1Š. To answer questions
on the differences g ÿ g 0 completely one needs to remove this restriction on the
test functions used in (2.3). For this Montgomery conjectured, on number-theoretic
grounds, that uniformly for jaj > 1 in bounded intervals,
F…a† ˆ 1 ‡ o…1†;
as T ! 1:
2:6†
Combining the conjecture (2.6) with (2.5), one can now use (2.3) to obtain two
important results on the zeros of the Riemann zeta-function. The ®rst is the pair
correlation conjecture
Z b
X
T
sin pu 2
log T
1,
1ÿ
du:
2:7†
N…T; b† ˆ
2p
pu
0
0
0 < g; g < T
0 < g ÿ g 0 < 2pb = log T
By (2.2) we see that the average spacing between zeros around height T is
2p= log T, and therefore N…T; b† counts the number of pairs of zeros within a
distance of b times the average. The derivation of (2.7) from (2.3), (2.5), and
(2.6) is not dif®cult and may be found in [11]. An immediate consequence of
(2.7) is that there are zeros very close together, namely
lim inf …gn ‡ 1 ÿ gn †
n!1
log gn
ˆ 0;
2p
2:8†
where gn denotes the nth zero ordered by height. The second important
consequence of (2.5) and (2.6) is that
X
T
mg ,
log T;
2:9†
N …T † ˆ
2p
0<g<T
where m g denotes the multiplicity of the zero r ˆ b ‡ ig, and the sum is over
zeros counting multiplicity. Letting Ns …T † denote the number of simple zeros with
0 < g < T, we see by the inequality
X
2 ÿ m g † < Ns …T † < N…T †;
2:10†
2N…T † ÿ N …T † ˆ
0<g<T
together with (2.2) and (2.9), that Ns …T † , N…T † , …T =2p† log T. Hence almost
all the zeros are simple. An interesting result of Gallagher and Mueller [10] is that
(2.7) by itself implies (2.9). (This is not dif®cult if one also assumes RH and uses
(2.5), but their result is unconditional.) Without the conjecture (2.6) one can only
prove partial results on the multiplicity and correlation of the zeros. Montgomery
34
d. a. goldston et al.
used (2.5) and special choices for r …u† in (2.3) to prove, assuming RH, that
N …T † < … 43 ‡ «†N…T †
and N …T † ‡ 2N…T; 0:6695 . . .† > N…T †
2:11†
for any « > 0 and T suf®ciently large. This implies that
lim inf
T !1
Ns …T † 2
>
and
N…T † 3
lim inf …gn ‡ 1 ÿ gn †
n!1
log gn
< 0:6695 . . . :
2p
2:12†
(Actually Montgomery did not compute the constant 0.6695 exactly and gave 0.68
as an upper bound.) The question of ®nding the best constants in (2.11) by using
the estimate (2.5) in (2.3) is an interesting extremal problem which is still
unsolved. At present the best constants known by this method are 0:67275 . . .
instead of 23 (see [4]) and 0:6072 . . . instead of 0:6695 . . . (see § 3). Since
Montgomery's work on these questions, other distinctly different approaches to
these problems have also been developed by Montgomery and Odlyzko [22] and
Conrey, Ghosh, and Gonek [6, 7]. In particular, the latter have shown that
lim inf
T !1
Ns …T † 19
>
ˆ 0:7073 . . . ;
N…T † 27
2:13†
log gn
< 0:5172 . . . :
lim inf …gn ‡ 1 ÿ gn †
n!1
2p
Both results are under the assumption of RH; the ®rst also assumes the
Generalized LindeloÈf Hypothesis for Dirichlet L-functions. It is interesting to
note that the second result holds for distinct zeros, while Montgomery's result for
small gaps between zeros in (2.12) might be due to the existence of a positive
proportion of multiple zeros. There are also results on large gaps between zeros
(see [6] and [22]). To complete this survey, Soundararajan [26], improving on
earlier work of Conrey, Ghosh, Goldston, Gonek, and Heath-Brown [5], has found
that on RH there must be a positive proportion of zeros with gap size less than
0.6878 times the average spacing.
3. Statement of results
We prove in this paper the following new result on F…a†.
Theorem. Assume the Generalized Riemann Hypothesis. Then for any « > 0
we have
3:1†
F…a† > 32 ÿ jaj ÿ «;
uniformly for 1 < jaj < 32 ÿ 2« and all T > T0 …«†.
Using the Theorem we can improve the estimates in (2.11) and (2.12) with the
assumption of GRH in place of RH.
Corollary. Assuming the Generalized Riemann Hypothesis we have
N …T † < …1:32611 . . .†N…T †
and
Ns > …0:67388 . . .†N…T †
3:2†
for all suf®ciently large T. Furthermore, if N …T † , …T =2p† log T, then we have
N…T; 0:57812 . . .† q N…T †;
3:3†
pair correlation of zeros
35
so that, in particular,
lim inf …gn ‡ 1 ÿ gn †
n!1
log gn
< 0:57812 . . . :
2p
3:4†
The result of Conrey, Ghosh, and Gonek in (2.13) is better for Ns …T †, but their
method does not give results on N …T †. Gallagher [9] has obtained on RH upper
and lower bounds for N…T; b† when b is an integer or half an integer, using (2.3)
and (2.5). Soundararajan's result mentioned above only gives (3.3) with 0.6878 on
RH; however (3.3) does not imply that a positive proportion of the zeros have
these short gaps since the result is for differences between zeros and is consistent
with a non-positive proportion of very close zeros.
We now prove the Corollary. Montgomery used the functions
sin pa 2
b
k…u† ˆ max…1 ÿ juj; 0†; k…a† ˆ
pa
in his proof of (2.11). We use instead the functions
8
< 1 ÿ juj ‡ sin 2pjuj if juj < 1,
2p
h…u† ˆ
:
0
if juj > 1;
2 sin pa
1
b
h…a† ˆ
:
pa
1 ÿ a2
Here b
h is the Selberg minorant for the characteristic function of the interval
‰ÿ1; 1Š with its Fourier transform h having support in ‰ÿ1; 1Š. This function was
used by Gallagher [9] in his work on N…T; b† mentioned above. We ®rst prove
(3.2). In (2.3) take r ˆ h. Then, since h is non-negative and h…0† ˆ 1, we have
Z 1
X
log T
T
b
w…g ÿ g 0 † ˆ
log T
h …g ÿ g 0 †
h…a†F…a† da:
N …T † <
2p
2p
ÿ1
0 < g; g 0 < T
Since b
h is non-positive for jaj > 1, we have by (2.4), (2.5), and the Theorem that
the right-hand side is at most
Z 3=2
Z 1
T
3
b
b
a h…a† da ‡ 2
2 ÿ a†h…a† da ‡ «
log T 1 ‡ 2
2p
0
1
T
ˆ …1 ‡ 0:336196708 . . . ÿ 0:010084208 . . . ‡ «†
log T
2p
T
log T;
ˆ …1:326112499 . . . ‡ «†
2p
where the last line is obtained by a numerical calculation. The lower bound for Ns
follows from this and (2.10).
To prove (3.3), we take r…u† ˆ b
h…u=l†, and note that this is a minorant for the
characteristic function of the interval ‰ÿl; lŠ. Thus
X
0 log T
b
w…g ÿ g 0 †
h …g ÿ g †
N …T † ‡ 2N…T; l† >
2pl
0
0 < g; g < T
Z 1=l
T
lh…la†F…a† da:
log T
ˆ
2p
ÿ1 = l
36
d. a. goldston et al.
Since the integrand is non-negative, we have by (2.2), (2.5), (3.1), and the
assumption N …T † , …T =2p† log T, that for any « > 0 and T suf®ciently large,
Z 1
Z 3=2
T
log T l ÿ 1 ‡ 2l
N…T; l† > … 12 ÿ «†
a h…la† da ‡ 2l
32 ÿ a†h…la† da :
2p
0
1
The second integral is the additional amount obtained by our Theorem. If we
ignore this integral, we ®nd by a numerical calculation that the right-hand side is
positive for l ˆ 0:6072 . . . and gives the result mentioned earlier on RH.
Including the last integral, we ®nd that the right-hand side is positive for
l ˆ 0:5781 . . . : This proves (3.3).
4. F…a† and mean values of Dirichlet series
Our ®rst step in proving the Theorem is to relate F…a† to the mean value of a
Dirichlet series over primes. This was done by Montgomery to obtain (2.5) and we
follow his approach initially. We will be using the results and notation from [13].
Throughout the rest of the paper we will use a more convenient form of the
function F…a†. We let
X
0
x i …g ÿ g † w…g ÿ g 0 †;
4:1†
F…x; T † ˆ
0 < g; g 0 < T
and thus have
F…a† ˆ
T
log T
2p
ÿ1
F…T a ; T †:
Equation (2.5) may be proved in the slightly stronger form (on RH)
s !!
T ÿ2 2
log log T
x log T ‡ log x† 1 ‡ O
F…x; T † ˆ
2p
log T
4:2†
4:3†
uniformly for 1 < x p T . This is Lemma 8 of [14]. (There the condition x < T
was given, but x p T holds with no change.) We are therefore interested in the
situation when
T p x;
4:4†
which we henceforth assume. This condition allows us to apply the results in [13].
Our starting point is an explicit formula of Montgomery [21]. Assuming RH
and x > 1, we have
X
xig
1 X L…n†
x3=2ÿit
ÿit
ˆ
ÿ 3
ÿ2x
2
x n < x nÿ1 = 2 ‡ i t
g 1 ‡ …t ÿ g†
2 ÿ it
X
L…n†
xÿ1 = 2 ÿ i t
log…jt j ‡ 2†
:
ÿ 1
‡O
‡x
3=2‡it
x
n>x n
2 ‡ it
4:5†
Let s ˆ j ‡ it, and let
A…s† ˆ
X L…n†
ns
n<x
and
A …s† ˆ
X L…n†
:
ns
n>x
4:6†
pair correlation of zeros
On writing
8Z
>
>
<
x
xs
0
ˆ
Z
>
s
>
:ÿ
u
sÿ1
1
x
Z
du ˆ
x
1
u
sÿ1
37
1
du ‡ O
; for Rs ˆ j > 0;
jsj
u s ÿ 1 du;
for Rs ˆ j < 0;
where the change in the ®rst integral's limit of integration is to agree with the
notation of [13], we can rewrite (4.5) as
Z x
X
xig
1
ÿit
1=2ÿit
1
A…ÿ 2 ‡ it † ÿ
ˆ
u
du
ÿ2x
2
x
1
g 1 ‡ …t ÿ g†
Z 1
3
ÿ3=2ÿit
u
du
‡ x A … 2 ‡ it † ÿ
x
log…jt j ‡ 2†
:
‡O
x
Montgomery proved that
2
xig
dt
2
1
‡
t
ÿ
g†
0<g<T
2
Z T X
2
xig
dt ‡ O…log3 T †:
ˆ
2
p 0
g 1 ‡ …t ÿ g†
2
F…x; T † ˆ
p
Z
4:7†
X
ÿ1
1
4:8†
This shows, in particular, that F…x; T † > 0 as stated in (2.2). Using the bound
X
1
p log…jt j ‡ 2†
2
g 1 ‡ …t ÿ g†
and (4.8), we have
Z T
X
xig
log…jt j ‡ 2† 2
ÿ 2xÿ i t
‡
O
dt
2
x
0
g 1 ‡ …t ÿ g†
T log2 T
‡ O…log3 T †;
ˆ 2pF…x; T † ‡ O
x
and thus we obtain, by (4.7) and (4.4),
Z x
Z T 1
1=2ÿit
1
du
2pF…x; T † ˆ
x A…ÿ 2 ‡ it † ÿ 1 u
0
2
Z 1
3
ÿ3=2ÿit
u
du dt ‡ O…log3 T †:
‡ x A … 2 ‡ it † ÿ
x
4:9†
We now introduce a smooth weight WU …t † as in [13]. We take WU …t † to have
support in ‰0; 1Š, 0 < WU …t † < 1, WU …t † ˆ 1 for 1=U < t < 1 ÿ 1=U , and
j†
WU …t † p U j for j ˆ 1; 2; 3 . . . : Here U ˆ log B T for some B > 1. We insert
WU …t =T † into the integral in (4.9) and extend the range of integration to all R. To
38
d. a. goldston et al.
bound the resulting change in the integral, we use the estimate
Z V ‡W Z x
1
1=2ÿit
1
du
x A…ÿ 2 ‡ it † ÿ 1 u
V
2
Z 1
3
ÿ3 = 2 ÿ i t
u
du dt
‡ x A … 2 ‡ it † ÿ
x
p F…x; V ‡ W † ÿ F…x; V † ‡ O…log3 …V ‡ W ††
X
w…g ÿ g 0 † ‡ O…log3 …V ‡ W ††
p
0< g0 <V ‡W
V <g<V ‡W
X
p
log jgj ‡ O…log3 …V ‡ W ††
V <g<V ‡W
p W log2 …V ‡ W † ‡ log3 …V ‡ W †;
for
P
W > 2 and V > 0. Here we have used the well-known estimate
1 p log t and the estimate below (4.8). We apply this with W ˆ T =U
and V ˆ 0 and V ˆ T ÿ T =U, and see that the weight introduces a change of at
most O……T =U † log2 T † in the right-hand side of (4.9). On multiplying out the
weighted integral, we see by Theorem 3 of [13] that the sum of the `cross' terms
is p x 1 ‡ « =T. Thus we have proved the following lemma.
t <g<t‡1
Lemma 1. Assume RH. If x q T and U ˆ log B T with B > 1, then we have
1‡« 1
x2
T log2 T
x
I2 …x; T † ‡ O
;
4:10†
I1 …x; T † ‡
‡O
F…x; T † ˆ
2
2p
T
U
2px
where
Z
I1 …x; T † ˆ
and
1
ÿ1
WU
2
Z x
t 1=2ÿit
1
dt
A…ÿ
‡
it
†
ÿ
u
du
2
T
1
2
Z 1
t 3
ÿ3=2ÿit
WU
u
du dt:
A … 2 ‡ it † ÿ
I2 …x; T † ˆ
T
ÿ1
x
Z
1
4:11†
4:12†
In order to evaluate the mean values above, one needs two types of information
concerning L…n†. First one requires a strong estimate for the sums of the squares
of the coef®cients L of A…s† and A …s†. Assuming RH, we see that the prime
number theorem takes the form
X
L…n† ˆ x ‡ O…x 1 = 2 log 2 x†;
4:13†
w…x† ˆ
n<x
so that by partial summation we have
X 2
L …n† ˆ x log x ÿ x ‡ O…x 1 = 2 log 3 x†:
n<x
4:14†
pair correlation of zeros
Second, one needs a good estimate for the number of prime twins. Let
8
Yp ÿ 1
>
>
if k is even, k 6ˆ 0,
2C
<
pÿ2
p
j
k
S…k† ˆ
p>2
>
>
:
0
if k is odd,
where
Y
1
1ÿ
;
Cˆ
p ÿ 1†2
p>2
39
4:15†
4:16†
and, for jk j < N an integer, let
N1 ˆ N1 …k† ˆ maxf0; ÿkg
and N2 ˆ N2 …k† ˆ minfN; N ÿ kg:
4:17†
Then a strong form of a conjecture of Hardy and Littlewood is that for 1 < jk j < N ,
X
L…n†L…n ‡ k† ˆ S…k†…N ÿ jkj† ‡ O…N 1 = 2 ‡ « †:
4:18†
N1 < n < N2
Assuming RH and this last conjecture, Bolanz [3] asymptotically evaluated I1
and I2 for T p x p T 3 = 2 ÿ « , and was able to extend this range to T p x p T 2 ÿ «
(written communication). Bolanz's argument is long and complicated, in part
because he does not use a smooth weight like WU . By using the results in [13]
together with some of the later argument in this paper, one can obtain the same
results in an easier fashion. It is possible to obtain them in an even easier fashion
by employing Parseval's theorem and Tauberian arguments as in [14]. The result
of any of these arguments is that, subject to RH and (4.18), F…x; T † , …T =2p† log T
for T < x p T 2 ÿ « .
In order to avoid the conjecture (4.18) we use an approximation to L…n†. This
again leads to mean values of Dirichlet series similar to those in Lemma 1, and
we evaluate them in the same way. The simplest approach of using the methods
of [14] is not available however, since we need to integrate the product of two
different Dirichlet series, and the non-negativity requirement for the Tauberian
argument is not met.
5. The lower bound method
As our approximation to L…n† we use
l Q …n† ˆ
X m2 …q† X
dm…d †:
f…q† d j q
q<Q
5:1†
djn
This function originated in work of Heath-Brown [16]. In place of (4.13) we have,
for 1 < Q < x,
X
l Q …n† ˆ x ‡ O…Q†:
5:2†
n<x
In place of (4.14), we have, for 1 < Q < x 1 = 2 ,
X 2
l Q …n† ˆ xL…Q† ‡ O…Q 2 †;
n<x
5:3†
40
d. a. goldston et al.
where
L…Q† ˆ
X m2 …q†
ˆ log Q ‡ O…1†;
f…q†
q<Q
5:4†
and instead of (4.18) we have for 1 < Q < x 1 = 2 , N ˆ ‰xŠ, and 1 < jk j < N,
X
kd…k†x
‡ O…Q 2 †:
l Q …n†l Q …n ‡ k† ˆ S…k†…x ÿ jk j† ‡ O
5:5†
f…k†Q
N <n<N
1
2
The proof of (5.2) is easy, and the proofs of (5.3) and (5.5) may be found in [12].
We thus see that l Q …n† satis®es a twin prime type conjecture like (4.18) for small
Q. Unfortunately (5.3) fails to give the same main term as in (4.14), and this is
the principal source of the loss in our lower bounds.
While l Q …n† is different from L…n† for individual n, its value as an
approximation to L…n† is roughly speaking that it behaves like L…n† on average. Let
X
L…n†;
5:6†
w…x; q; a† ˆ
n<x
n a…q†
and let Ea; b equal 1 if …a; b† ˆ 1 and equal zero otherwise. Let
x
:
E…x; q; a† ˆ w…x; q; a† ÿ Ea; q
f…q†
Then for 1 < Q < x we have
X
l Q …n†L…n† ˆ w…x†L…Q† ‡ O…Q log x†;
5:7†
5:8†
n<x
and for 1 < jk j < N ˆ ‰xŠ we have
X
l Q …n†L…n ‡ k†
N1 < n < N2
kd…k†x
ˆ S…k†…x ÿ jk j† ‡ O
f…k†Q
X 2
m …q†q log…2Q=q†
jE…N2 ‡ k ; q; k† ÿ E…N1 ‡ k ; q; k†j :
‡O
f…q†
q<Q
5:9†
These results are proved in [12]. Assuming GRH the prime number theorem for
arithmetic progressions takes the form
E…x; q; a† p x 1 = 2 log2 …qx†;
1=2‡«
5:10†
:
so the second error term in (5.9) is p Qx
By exploiting the similarity between L…n† and l Q …n† and using a Bessel
inequality type argument, we are able to extract some of the information
contained in the unproved conjecture (4.18). Denote by A Q the same Dirichlet
series as A, but with l Q …n† in place of L…n†. Then we have
2
Z 1
t 1
1
dt > 0:
WU
‡
it
†
ÿ
A
ÿ
‡
it
†
5:11†
A…ÿ
Q
2
2
T
ÿ1
R
The expected size of both Dirichlet series is 1x u 1 = 2 ÿ i t du because of (4.13) and
pair correlation of zeros
41
(5.2). We may thus subtract this factor from both series in (5.11) and multiply out
to obtain
Z 1
Z x
t
1=2ÿit
1
A…ÿ 2 ‡ it † ÿ
WU
u
du
I1 …x; T † > 2R
T
ÿ1
1
Z x
u 1 = 2 ÿ i t du dt
´ A Q …ÿ 12 ‡ it † ÿ
1
2
Z x
t 1=2ÿit
1
WU
u
du dt:
A Q …ÿ 2 ‡ it † ÿ
ÿ
T
ÿ1
1
Z
1
5:12†
We now apply Corollary 1 of [13] with j ˆ ÿ 12 , and obtain, for T p x p T 2 ÿ « ,
X
X 2
b U …0†T 2
L…n†l Q …n†n ÿ
l Q …n†n
I1 …x; T † > W
n<x
n<x
3 Z 1 X
T
2
b U …v† dv
S…h†h RW
‡ 4p
2p
v3
T = 2p x
h < 2p x v = T
Z 2p x v = T
3 Z 1
T
2
b U …v† dv ‡ R1 …x; T †;
ÿ 4p
u du RW
2p
v3
T = 2p t x
0
5:13†
where
t ˆ T 1ÿ«
5:14†
and R1 …x; T † denotes the sum of the error terms appearing in Corollary 1 of [13].
We shall deal with R1 …x; T † in the next section.
The lower bound for I2 …x; T † is obtained in exactly the same manner, and
Corollary 2 of [13] then gives (with j ˆ 32 , h ˆ 12 ÿ «, l ˆ 2), for T p x p T 3 = 2 ÿ « ,
X
L…n†l Q …n† X l2Q …n†
b U …0†T 2
ÿ
I2 …x; T † > W
n3
n3
n>x
n>x
Z
8p 2 T = 2p x X S…h†
b U …v†v dv
RW
‡
T 0
h2
h<H
Z
X
8p2 T H = 2p x
S…h†
b U …v†v dv
RW
‡
2
T T = 2p x
h
2px v = T < h < H
8p2
ÿ
T
Z
T H = 2p x
0
Z
H
2px v = T
du
b U …v†v dv ‡ R2 …x; T †;
RW
u2
5:15†
where R2 …x; T † denotes the error terms in Corollary 2, and
H ˆ …x=T †2 …xT †« :
5:16†
6. The error terms
We consider ®rst R1 …x; T †. By Corollary 1 of [13] we have
R1 …x; T † p …x 3 ‡ « =T † ‡ x 2 ‡ max…v; f† ‡ « ‡ x« ;
6:1†
42
d. a. goldston et al.
where v and f will be explained below. Now xv is the maximum of the error
terms in (4.13) and (5.2). We will always take
Q ˆ x n;
0 < n < 12 ;
1=2‡«
6:2†
1
2
so that these error terms are p x
. Thus we have v ˆ ‡ «. The number f is
determined from the error terms in (5.5) and (5.9). We will assume in what
follows that
6:3†
0 < k < x h ; 0 < h < 12 ÿ «:
First, using l Q …n† p n« and S…k† p log log k, we have, by (5.5),
X
k d…k†x
«
‡ O…Q 2 †
l Q …n†l Q …n ‡ k† ˆ S…k†x ‡ O…k x † ‡ O
f…k†Q
n<x
ˆ S…k†x ‡ O…x 1 ÿ n ‡ « † ‡ O…x 2 n †:
6:4†
Similarly by (6.2) and (5.9) we have
X
l Q …n†L…n ‡ k†
n<x
ˆ S…k†x ‡ O…k x « † ‡ O…x 1 ÿ n ‡ « †
X 2
m …q†q log…2Q=q†
jE…N2 ‡ k ; q; k† ÿ E…N1 ‡ k ; q; k†j :
‡O
f…q†
q<Q
Since
6:5†
k log x
;
jE…N2 ‡ k ; q; k† ÿ E…N1 ‡ k ; q; k†j ˆ jE…x; q; k†j ‡ O
f…q†
this gives
X
X
l Q …n†L…n ‡ k† ˆ S…k†x ‡ O…x 1 ÿ n ‡ « † ‡ O log2 Q
jE…x; q; k†j : …6:6†
n<x
q<Q
The number f in (6.1) is any number such that the error terms in (6.4) and (6.6)
are p xf . Using the GRH estimate (5.10), we see that the last error term in (6.6)
is p Qx 1 = 2 ‡ « . Thus we can take f ˆ max…1 ÿ n; 2n; n ‡ 12 † ‡ «. However, if we
estimate this last error term while averaging over k, we can do better. For this we
need a more precise expression for R1 …x; T † than that in (6.1). We use the error
term from Corollary 1 of [13] estimated above except for the last error term in
(6.6); for it we use the expression in the proof of Corollary 1 of [13] which gave
rise to this error term. Setting
X
jE…x; q; k†j;
6:7†
GQ …x; k† ˆ
q<Q
in place of (6.1) we obtain
3
x
‡ x 3 ÿ n ‡ x 2 ‡ 2n ‡ T
R1 …x; T † p
T
X
1<k<H
Z
xÿk
ktÿ
K…u; k† d u GQ …u; k† x« ;
6:8†
where
x < t 1 = …1 ÿ h† ;
H ˆ x=…t ‡ 1†;
t ˆ T 1 ÿ «;
6:9†
pair correlation of zeros
43
and K…u; k† ˆ K ÿ 1 = 2 …u; k† (see [13]) is a smooth function such that
¶
K…u; k† p T « ; for k < u=t:
6:10†
¶u
To estimate the term including GQ …x; k† we use Hooley's estimate (see [18])
X
max jE…v; q; a†j2 p u log4 2u;
K…u; k† p u;
v<u
1<a<q
a; q† ˆ 1
which assumes GRH. (This bound without the max is a well-known result of
TuraÂn and of Montgomery [19].) By Cauchy's inequality we ®nd that
X
max GQ …v; k†
k<y
v<u
ˆ
X X
max jE…v; q; k†j
v<u
q<Q k<y
ˆ
p
X
max jE…v; q; k†j ‡ O
log q
X
X
q<Q
k<y
k; q† ˆ 1
X
y
1=2
py
max jE…v; q; k†j
k<y
k; q† ˆ 1
X
q<Q
p y1=2
q<Q
X
q<Q
1=2
v<u
y
1‡
q
2
v<u
1 = 2 X
1<a<q
a; q† ˆ 1
1 = 2
‡ O…Q log Q†
max jE…v; q; a†j
v<u
2
1 = 2
‡ O…Q log Q†
X 1 = 2 y
1‡
u 1 = 2 log2 …2u† ‡ O…Q log Q†
q
q<Q
< …y 1 = 2 Q ‡ yQ 1 = 2 †u 1 = 2 log2 …Qu†:
6:11†
On integrating by parts and using (6.9), (6.10), and (6.11), we obtain
X Z xÿk
K…u; k† d u GQ …u; k†
T
1<k<H
ktÿ
X jK…x ÿ k; k†GQ …x ÿ k; k†j ‡ jK…kt; k†GQ …ktÿ ; k†j
pT
1<k<H
¶K…u; k† ‡
¶u GQ …u; k† du
ktÿ
X
Z xÿk X
1‡«
max GQ …u; k† ‡ T
GQ …u; k† du
p xT
Z
1<k<H
px
3=2‡«
T
1‡«
xÿk
u<x
H
1=2
ktÿ
Q ‡ HQ
1=2
p …x 2 ‡ n T 1 = 2 ‡ x 5 = 2 ‡ n = 2 †…xT †« :
1<k<H
†
6:12†
44
d. a. goldston et al.
We conclude that for any « > 0, n < 12 , and T p x < T 3 = 2 ,
x
x1ÿn x2n
xn
x1=2‡n=2
2
‡
‡ 1=2 ‡
T «:
‡
R1 …x; T † p x T
T
T
T
T2
T
6:13†
We choose Q ˆ x n as large as possible while still keeping R1 …x; T † p x 2 T. We
thus take
6:14†
Q ˆ x n ˆ T 1 = 2 ÿ « and x < T 3 = 2 ÿ 2« ;
where « is the same as in (6.13), and ®nd that with these restrictions
R1 …x; T † p x 2 T:
6:15†
By Corollary 2 of [13] and an argument similar to that for R1 …x; t †, we ®nd that
ÿ1
X Z 1
x
ÿ1 ÿ n
2nÿ2
‡x
R2 …x; T † p
‡x
‡T
J…u; k† d u GQ …u; k† x « ;
T
max…x;
k
t†
1<k<H
6:16†
where
x 2 = …1 ÿ «† x 2
p …xT †« ; t ˆ T 1 ÿ « :
T
t2
The function J…u; k† is a smooth function satisfying the following estimates
(see [13]):
H ˆ
¶J…u; k†
u
p uÿ 4 T « ; for k < :
¶u
t
We note, using the bounds (6.11) and (6.17), that
Z 1
Z 1
¶J…u; k†
J…u; k† d u GQ …u; k† ˆ ÿJ…v; k†GQ …v; k† ÿ
GQ …u; k† du
¶u
v
v
Z 1
1
du
GQ …u; k† 4 :
< 3 GQ …v; k† ‡ T «
v
u
v
We now consider
X Z 1
J…u; k† d u GQ …u; k†
J…u; k† ˆ K3 = 2 …u; k† p uÿ3 ;
1<k<H
ˆ
max…x; k t†
X
1<k <x=t
Z
x
1
J…u; k† d u GQ …u; k† ‡
X
Z
x=t<k <H 1
kt
6:17†
6:18†
J…u; k† d u GQ …u; k†
ˆ S1 ‡ S2 ;
6:19†
say. By (6.11), (6.17), and (6.18) we have
Z 1 X
1 X
du
GQ …x; k† ‡ T «
GQ …u; k† 4
S1 p 3
x k <x=t
u
x
k <x=t
1 = 2
x
x 1=2 1=2‡«
Q‡ Q
x
t
t
Q
Q1=2
«
:
p …xT †
‡
T 1 = 2 x 2 Tx 3 = 2
T«
p 3
x
6:20†
pair correlation of zeros
45
P
1 We break S2 into sums of the form
K < k < 2 K , where x=t < K < 2 H , and obtain
X Z 1
J…u; k† d u GQ …u; k†
K <k <2K
kt
1
p
Kt†3
p
p
X
K <k <2K
GQ …kt; k† ‡ T
«
K <k <2K
X
1
max GQ …v; k† ‡ T «
3
v
Kt† k < 2 K < 2 Kt
K 1 = 2 Q ‡ KQ 1 = 2 †T «
Kt†5 = 2
Z
X
Z
1
Kt
1
kt
GQ …u; k†
du
u4
GQ …u; k†
du
u4
X
k <2K
:
Letting K ˆ ‰x=tŠ; 2‰x=tŠ; . . . ; we see that the estimate (6.20) also holds for S2 .
We therefore have, by (6.16),
T
x
x1ÿn x2n
Q
x 1 = 2Q 1 = 2
‡
‡ 1=2 ‡
xT †« :
‡
6:21†
R2 …x; T † p 2
T
T
T
x
T2
T
The term in parentheses is the same as in (6.13) and therefore, subject to (6.14),
we have
6:22†
R2 …x; T † p T =x 2 :
7. Proof of the Theorem
To prove the Theorem we will show that
and
I1 …x; T † > 12 Tx 2 log…TQ=x† ‡ O…Tx 2 …log x†2 = 3 log log T †
7:1†
T
T
2=3
I2 …x; T † > 2 log…TQ=x† ‡ O 2 …log x† log log T
2x
x
7:2†
for Q ˆ T 1 = 2 ÿ « and T p x < T 3 = 2 ÿ 2« . The Theorem then follows immediately
from Lemma 1. The error terms in (7.1) and (7.2) may be reduced with more
effort but this improvement is not needed here.
We assume from now on that Q and x satisfy (6.14). We ®rst consider I1 …x; T †.
By partial summation, using (5.3) and (5.8), we have
X 2
X
L…n†lQ …n†n ÿ
lQ …n†n ˆ x 2 log Q ‡ O…x 2 †:
2
n<x
n<x
In order to handle WU …v† we use the following estimates from [13]:
b U …0† ‡ O…v†;
b U …v† ˆ W
W
b U …v† p min…1; U =v†;
W
7:3†
where U ˆ …log T †B , with B > 1. Next, using (7.3) we note that in the last
integral in (5.13) we may change the ®rst lower limit of integration from T =2ptx
to T =2px with an error that is
Z T = 2p x 3
x
dv p Tx 2 :
pT3
T
T = 2p t x
46
d. a. goldston et al.
Therefore we ®nd by (5.13) and (6.15) that
b U …0†Tx 2 log Q
I1 …x; T † > W
3 Z 1 X
Z 2p x v = T
T
2
2
b U …v† dv
S…h†h ÿ
u du RW
‡ 4p
2p
v3
T = 2p x
0
h < 2p x v = T
‡ O…Tx 2 †
b U …0†Tx 2 log Q ‡ I11 …x; T † ‡ O…Tx 2 †;
>W
say. By a change of variable,
Z 1X
y3
yT
dy
2
2
bU
RW
S…h†h ÿ
I11 ˆ 2Tx
2px y 3
3
1
h<y
Z 1
b U yT dy ;
ˆ 2Tx 2
S2 …y†RW
2px y 3
1
where we set
X
ya‡1
:
S…h†ha ÿ
Sa …y† ˆ
a‡1
h<y
7:4†
7:5†
In order to evaluate this integral we need a result on averages of S…h†, namely that
S0 …y† ˆ ÿ 12 log y ‡ O……log y†2 = 3 †
7:6†
(see [8]). From this result and by partial summation, we see that for a > 0,
Sa …y† p y a log 2 = 3 y:
7:7†
By (7.5), (7.6) and (7.7) we obtain
Z W
Z W
X
dy
dy 1
S2 …y† 3 ˆ
S…h†h2
ÿ …W ÿ 1†
y
y3 3
1
h
h<W
ˆ 12 S0 …W † ÿ
1
S2 …W † ‡ 13
2W 2
ˆ ÿ 14 log W ‡ O…log 2 = 3 W †:
7:8†
Now by (7.3), (7.7), and (7.8) we have
Z x=T
dy
b U …0† ‡ O yT
S2 …y† W
I11 ˆ 2Tx 2
2px
y3
1
Z xU = T
Z 1
U
dy
2
2
2 = 3 dy
2
2
2=3
‡ O Tx
y …log y†
y …log y†
‡ O Tx
yT =2px y 3
y3
x=T
xU = T
Z x=T
dy
2
b
S2 …y† 3 ‡ O…Tx 2 …log x†2 = 3 log U †
ˆ 2WU …0†xT
y
1
b U …0† log…T =x† ‡ O…Tx 2 …log x†2 = 3 log log T †:
ˆ 12 Tx 2 W
b U …0† ˆ 1 ‡ O…1=U † (see [13]).
Inequality (7.1) now follows on using W
pair correlation of zeros
47
We now treat the various terms in (5.15) to obtain a lower bound for I2 …x; T †.
By partial summation using (5.3) and (5.8), we ®nd that
X L…n†l Q …n† X l2Q …n† 1 log Q
1
2
ÿ
ˆ
‡O 2 :
3
3
2
2 x
n
n
x
n>x
n>x
Next, by (7.3),
Z T = 2p x
Z T = 2p x X
8p2
S…h†
T
b U …v†v dv p 1
v dv p 2 ;
R
W
2
T 0
T
h
x
0
h<H
7:9†
since by (7.6) the sum over h, extended to in®nity, converges. In the second
integral in (5.15) we may replace the lower limit of integration by 0 with an error
less than the expression on the left-hand side in (7.9). On making the change of
variable v ˆ 2pxy=T, we conclude that
b U …0†T log Q
I2 …x; T † > W
2x 2
Z H Z H X
2T
S…h†
du
b U Ty y dy ‡ O T
‡ 2
ÿ
R
W
2px
x
h2
u2
x2
1
y
y<h<H
b U …0†T log Q ‡ I21 …x; T † ‡ O T ;
7:10†
ˆW
2x 2
x2
say. By partial summation using (7.6), we have for a > 1,
X S…h†
1
log 2 = 3 y
ˆ
‡O
;
Ta …y† ˆ
ha
ya
a ÿ 1†y a ÿ 1
h>y
so that
Z
W
1
7:11†
Z
1
X
S…h† min…h; W †
T2 …y†y dy ˆ
y dy
h2 1
hˆ1
ˆ 12 S0 …W † ‡ 12 W ‡ 12 W 2 T2 …W † ‡ O…1†
ˆ W ÿ 14 log W ‡ O……log W †2 = 3 †:
7:12†
(With more effort one can prove (7.8) and (7.12) in a different way with an error
term of O…1†.) Returning to I21 , we note by (7.11) that
X S…h† Z H du
1
1
ÿ
ˆ T2 …y† ÿ T2 …H † ÿ ‡ 2
2
y H
h
u
y
y<h<H
1
log H †2 = 3
ˆ T2 …y† ÿ ‡ O
y
H †2
and therefore, by (7.3),
Z 2T H
1
log H †2 = 3
b U Ty y dy
T2 …y† ÿ ‡ O
R
W
I21 ˆ 2
y
2px
x 1
H †2
Z H 2T
1
Ty
2T…log x†2 = 3
b
RWU
y dy ‡ O
T2 …y† ÿ
:
7:13†
ˆ 2
y
2px
x 1
x2
48
d. a. goldston et al.
Now by (7.3), (7.11), and (7.12) we have
Z H 1
Ty
b
RWU
y dy
T2 …y† ÿ
y
2px
1
Z x=T 1
Ty
b
WU …0† ‡ O
y dy
T2 …y† ÿ
ˆ
y
x
1
Z H
Z xU = T
log y†2 = 3
log y†2 = 3 xU
y dy
y dy ‡ O
‡O
Ty
y2
y2
x=T
xU = T
x 2=3
b U …0† ÿ 1 log x ‡ O
log
ˆW
‡ O……log x†2 = 3 log U †
4
T
T
ˆ ÿ 14 log…x=T † ‡ O……log x†2 = 3 log log T †:
7:14†
Inequality (7.2) now follows from (7.10), (7.13) and (7.14). This completes the
proof of the Theorem.
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D. A. Goldston
Department of Mathematics
and Computer Science
San Jose State University
San Jose
CA 95192
USA
goldston@jupiter.sjsu.edu
S. M. Gonek
Department of Mathematics
University of Rochester
Rochester
NY 14627
USA
gonek@math.rochester.edu
È zluÈk and C. Snyder
A. E. O
Department of Mathematics and Statistics
University of Maine
Orono
ME 04469
USA
and
Research Institute of Mathematics
Orono
ozluk@gauss.umemat.maine.edu
snyder@gauss.umemat.maine.edu