ANNALS OF
MATHEMATICS
Moments of the Riemann zeta function
By Kannan Soundararajan
SECOND SERIES, VOL. 170, NO. 2
September, 2009
anmaah
Annals of Mathematics, 170 (2009), 981–993
Moments of the Riemann zeta function
By K ANNAN S OUNDARARAJAN
In memoriam Atle Selberg
Abstract
Assuming the Riemann hypothesis, we obtain an upper bound for the moments
of the Riemann zeta function on the critical line. Our bound is nearly as sharp as
the conjectured asymptotic formulae for these moments. The method extends to
moments in other families of L-functions.
1. Introduction
An important problem in analytic number theory is to gain an understanding
of the moments
Z
T
Mk .T / D
0
j. 21 C i t /j2k dt:
2
For positive real numbers k, it is believed that Mk .T / Ck T .log T /k for a positive
constant Ck . A precise value for Ck was conjectured by Keating and Snaith [11]
based on considerations from random matrix theory. Subsequently, an alternative
approach, based on multiple Dirichlet series and producing the same conjecture,
was given by Diaconu, Goldfeld and Hoffstein [4]. Recent work by Conrey et al
[1] gives a more precise conjecture, identifying lower order terms in an asymptotic
expansion for Mk .T /.
Despite many attempts, asymptotic formulae for Mk .T / have been established
only for k D 1 (due to Hardy and Littlewood; see [22]) and k D 2 (due to Ingham;
2
see [22]). However we do have the lower bound Mk .T / k T .log T /k . This was
established by Ramachandra [13] for positive integers 2k, by Heath-Brown [6] for
all positive rational numbers k, and, assuming the truth of the Riemann hypothesis
(RH), by Ramachandra [12] for all positive real numbers k. See also the elegant
note [2] giving such a bound assuming RH, and [20] for the best known constants
The author is partially supported by the National Science Foundation and the American Institute of
Mathematics (AIM).
981
982
KANNAN SOUNDARARAJAN
implicit in these lower bounds. Analogous conjectures exist (see [1], [4], [10]) for
moments of central values of L-functions in families, and in many cases lower
bounds of the conjectured order are known (see [16] and [17]).
Here we study the problem of obtaining upper bounds for Mk .T /. When
0 k 2, Ramachandra, in [13] and [14], and Heath-Brown, in [6] and [7], showed,
2
assuming RH, that Mk .T / T .log T /k . The Lindelöf hypothesis is equivalent to
the estimate Mk .T / k;" T 1C" for all natural numbers k. Thus, for k larger than
2, it seems difficult to make unconditional progress on bounding Mk .T /. If we
assume RH, then a classical bound of Littlewood (see [22]) gives that (for t 10
and some positive constant C )
log t
(1)
j. 21 C i t /j exp C
;
log log t
and therefore Mk .T / T exp.2kC log T = log log T /. We improve upon this,
obtaining an upper bound of nearly the conjectured order of magnitude.
C OROLLARY A. Assume RH. For every positive real number k, and every
" > 0, we have
2
T .log T /k k Mk .T / k;" T .log T /k
2 C"
:
Our proof suggests that the dominant contribution to the 2k-th moment comes
from t such that j. 21 C i t /j has size .log T /k , and this set has measure about
2
T =.log T /k .
C OROLLARY B. Assume RH. Let k 0 be a fixed real number. For large T
we have
measft 2 Œ0; T  W j. 12 C i t /j .log T /k g D T .log T /
k 2 Co.1/
:
We will deduce these corollaries by finding an upper bound on the frequency
with which large values of j. 21 C i t /j can occur. Throughout we define
S.T; V / D ft 2 ŒT; 2T  W logj. 12 C i t /j V g;
and observe that
Z 2T
(2)
T
j. 21
2k
Z
C i t /j dt D
Z
D 2k
1
e 2kV d meas.S.T; V //
1
1
e 2kV meas.S.T; V //d V:
1
To prove Corollaries A and B, we desire estimates for the measure of S.T; V / for
large T and all V 3. To place our main theorem in context, let us recall the beautiful
result of Selberg that as t varies in ŒT; 2T , the distribution of logj. 12 C i t /j is
approximately Gaussian with mean 0 and variance 12 log log T . Precisely, Selberg’s
983
MOMENTS OF THE RIEMANN ZETA FUNCTION
theorem (see [18] and [19]) shows that for any fixed 2 R, and as T ! 1,
Z 1
q
2
1
1
meas S.T; 2 log log T / D T p
e x =2 dx C o.1/ :
2 p
Although Selberg’s result holds only for V of size log log T , we may speculate
that in a much larger range for V a similar estimate holds:
p
log log T
V2
(3)
meas.S.T; V // T
exp
:
log log T
V
2
Such an estimate would lead, via (2), to the bound Mk .T / k T .log T /k . In
our main theorem we establish that a weaker form of (3) holds in the range 3 V D o..log log T / log3 T /, where throughout we write log3 for log log log. In the
application to moments, the crucial range is when V is of size about k log log T ,
and our theorem shows that a version of (3) holds for such V . For larger values
of V , we obtain an upper bound of the form
meas.S.T; V // T exp.
1
129 V
log V /
(at least when 12 .log log T / log3 T < V ), and the shape of this estimate is in keeping
with Littlewood’s bound (1).
T HEOREM. Assume RH. Let p
T be large, let V 3 be a real number, and let
log log T V log log T , then
4
V
V2
1
I
meas.S.T; V // T p
exp
log log T
log3 T
log log T
S.T; V / be as defined above. If 10
if log log T < V 12 .log log T / log3 T , then
V
V2
meas.S.T; V // T p
exp
1
log log T
log log T
7V
4.log log T / log3 T
2 I
and, finally, if 12 .log log T / log3 T < V , then
meas.S.T; V // T exp
1
129 V
log V :
In the limited range 0 V log log T , Jutila [8] had previously shown that
V
V2
meas.S.T; V // T exp
1CO
:
log log T
log log T
p
I have shown recently in [21] that in the range 3 V 15 log T = log log T one
has
T
V2
meas.S.T; V // exp 10
:
log T
.log T /4
log 8V 2 log V
In contrast to our theorem above, these results are unconditional.
984
KANNAN SOUNDARARAJAN
As mentioned already, we build on Selberg’s work on the distribution of
log . 12 C i t /. Selberg computed the moments of the real and imaginary parts of
log . 12 C i t /. To achieve this, he found an ingenious expression for log . 21 C i t /
in terms of primes. His ideas work very well for the imaginary part of the logarithm,
but are more complicated for the real part of the logarithm owing to zeros of the
zeta function lying very close to 12 C i t. One novelty in our work is the realization
that if we seek an upper bound for logj. 12 C i t /j (which is what is needed for our
theorem), then the effect of the zeros very near 12 C i t is actually benign. This is
our main proposition given below, from which we will deduce our theorem. We
should comment that Selberg’s work is unconditional, and uses zero-density results
which put most of the zeros near the critical line. It would be interesting to see how
much of our work can be recovered unconditionally.
P ROPOSITION. Assume RH. Suppose T is large, let t 2 ŒT; 2T , and let
2 x T 2 . Let 0 D 0:4912 : : : denote the unique positive real number satisfying
e 0 D 0 C 20 =2. For all 0 , we have the estimate
X
log.x=n/ .1 C / log T
ƒ.n/
1
:
logj. 12 Ci t /j Re
C
CO
1
log x
log x
2
log x
C Ci t
nx n 2 log x
log n
Taking x D .log T /2 " in our proposition, and estimating the sum over n
trivially, we obtain the following explicit form of Littlewood’s bound (1), which improves upon the previous estimate obtained by Ramachandra and Sankaranarayanan
[15]. There is certainly some scope to improve our Corollary C below, and it may be
instructive to understand what the limit of the method would be (in a way analogous
to the elegant treatment of Im log . 12 C i t / given by Goldston and Gonek [5]).
C OROLLARY C. Assume RH. For all large t, we have
1C0
log t
3 log t
j. 21 C i t /j exp
C o.1/
exp
:
4
log log t
8 log log t
The method developed here is robust and applies equally well to moments in
families of L-functions; we discuss this briefly in Section 4 below. We end the
introduction by deriving Corollaries A and B.
Proof of Corollaries A and B. As mentioned earlier, the lower bound for
Mk .T / in Corollary A is due to Ramachandra [12]. The upper bound follows upon
inserting the bounds of our theorem into (2). In performing this computation, it is
convenient to use our theorem in the crude form
meas.S.T; V // T .log T /o.1/ exp. V 2 = log log T / for 3 V 4k log log T ,
meas.S.T; V // T .log T /o.1/ exp. 4kV /
for V > 4k log log T .
985
MOMENTS OF THE RIEMANN ZETA FUNCTION
From our theorem, we may see that the contribution to Mk .T / from those
t 2 Œ0; T  with j. 21 C i t /j > .log T /kC" or with j. 12 C i t /j < .log T /k " is
2
2
o.T .log T /k /. Combining this with the lower bound Mk .T / k T .log T /k we
obtain the lower bound for the measure implicit in Corollary B. The upper bound
implicit there follows from our theorem.
2. Proof of the main proposition
In proving our proposition we may suppose that t does not coincide with the
ordinate of a zero of .s/. Letting D 12 C i run over the nontrivial zeros of .s/,
we define
X 1
X
1=2
F .s/ D Re
D
:
s . 1=2/2 C.t /2
Visibly F .s/ is nonnegative in the half-plane 1=2. Recall Hadamard’s factorization formula which gives (see [3, (8) and (11) of Ch. 12])
Re
0
.s/ D
Re
1
s 1
C 21 log 1
2
Re
€0 1
. s C 1/ C F .s/;
€ 2
so that for t 2 ŒT; 2T  an application of Stirling’s formula yields
(4)
Re
0
.s/ D
1
2
log T C O.1/
Integrating (4) as D Re.s/ varies from
0 C i t ,
logj. 21 C i t /j
D
1
2
F .s/:
to 0 .> 12 /, we obtain, setting s0 D
logj.s0 /j
1
2
log T C O.1/ .0
D .0
1 1
2/ 2
1
2/
log T C O.1/
Z
0
1=2
F . C i t /d
1 2
2 / C .t
.t /2
.0
1X
log
2
/2
:
Since log.1 C x 2 / x 2 =.1 C x 2 /, we deduce that
(5)
logj. 21 C i t /j
logj.s0 /j .0
1 1
2/ 2
log T C O.1/
1
2 F .s0 /
:
L EMMA 1. Unconditionally, for any s not coinciding with 1 or a zero of .s/
and for any x 2, we have
0 0
X ƒ.n/ log.x=n/
0
1
1 X x s
.s/ D
C
.s/
C
log x log x
. s/2
ns
log x
nx
1
x1 s
1 X x 2k s
C
:
.1 s/2 log x log x
.2k Cs/2
kD1
986
KANNAN SOUNDARARAJAN
Proof. This is similar to an identity of Selberg; see Titchmarsh [22, Th. 14.20].
With c D max.1; 2 /, we consider
Z cCi 1
X ƒ.n/
1
0
xw
.s C w/ 2 dw D
log.x=n/;
2 i c i 1
w
ns
nx
which follows from integrating term by term using the Dirichlet series expansion of
. 0 =/.s C w/. On the other hand, moving the line of integration to the left and
calculating residues, this equals
0
.s/ log x
X x s
x1 s
C
2
. s/
.1 s/2
0
0
.s/
1
X
x 2k s
:
.2k Cs/2
kD1
Equating these two expressions we obtain the lemma.
Take s D C i t in Lemma 1, extract the real parts of both sides, and integrate
over from 0 to 1. Thus, for 2 x T 2 ,
X
ƒ.n/ log.x=n/
1 0
logj.s0 /j D Re
.s /
log x 0
ns0 log n log x
nx
Z 1 s
1 X
1
x
C
d
C
O
:
2
log x
log x
0 . s/
Using (4), and observing that
ˇ X Z 1 1=2 XˇˇZ 1 x s
ˇ
x
d ˇ d
ˇ
2
2
.
s/
js
0 j
0
0
D
X
x 1=2 0
x 1=2 0 F .s0 /
D
;
.0 1=2/ log x
js0 j2 log x
we deduce that
(6)
logj.s0 /j Re
X
nx
ƒ.n/ log.x=n/
log n log x
ns0
C
F .s0 /
x 1=2 0 F .s0 /
1
C
O
C
:
log x
log x
.0 1=2/ log2 x
log T
2 log x
Adding the inequalities (5) and (6), we obtain that
X ƒ.n/ log x=n
1
(7) logj. 21 C i t /j Re
C
log
T
0
2
ns0 log n log x
nx
C F .s0 /
x 1=2
.0
0
1=2/ log2 x
1
log x
1
2 .0
1
2
C
1
2/
1
log x
CO
1
:
log x
MOMENTS OF THE RIEMANN ZETA FUNCTION
987
We choose 0 D 12 C =log x, where 0 . This restriction on ensures that the
term involving F .s0 / in (7) makes a negative contribution, and may therefore be
omitted. The proposition follows.
3. Proof of the theorem
Our proof of the theorem rests upon our main proposition. We begin by
showing that the sum over prime powers appearing there may be restricted just to
primes.
L EMMA 2. Assume RH. Let T t 2T , let 2 x T 2 , and let 12 . Then
ˇ
ˇ
ˇX
ƒ.n/ log x=n ˇˇ
ˇ
log log log T C O.1/:
ˇ
n Ci t log n log x ˇ
nx
n¤p
Proof. The terms n D p k for k 3 clearly contribute an amount 1, and
it remains to handle the terms n D p 2 . By following closely the explicit formula
proof of the prime number theorem (see [3, 17 and 18]) we obtain that, on RH,
X
p
.log p/p 2i t z=T C z.log zT /2 :
pz
By partial summation, using this estimate when z .log T /4 and the trivial z
for smaller z, we deduce that for 12
p
X
1
log. x=p/
log log log T:
p
2C2i t
log x
p p
p x
We also need a standard mean value estimate whose proof we include for
completeness.
L EMMA 3. Let T be large, and let 2 x T . Let k be a natural number such
that x k T = log T . For any complex numbers a.p/ we have
Z 2T ˇ X
X ja.p/j2 k
a.p/ ˇˇ2k
ˇ
dt
T
kŠ
:
ˇ
ˇ
p
p 1=2Ci t
T
px
px
Proof. Write
X a.p/ k
X ak;x .n/
D
;
1=2Ci t
1=2Ci t
p
n
k
px
nx
where ak;x .n/ D 0 unless n is the product of k (not necessarily distinct) primes, all
Q
below x. In that case, if we write the prime factorization of n as n D riD1 pi˛i ,
988
KANNAN SOUNDARARAJAN
Qr
k
˛i
i D1 a.pi / .
˛1 ;:::;˛r
then ak;x .n/ D
Z
2T ˇ X
ˇ
ˇ
T
px
p
Now
Z
ak;x .m/ak;x .n/ 2T n i t
dt
p
m
mn
T
k
m;nx
X
X jak;x .n/j2
jak;x .m/ak;x .n/j
CO
DT
p
n
mn jlog.m=n/j
k
k
X
a.p/ ˇˇ2k
ˇ dt D
1=2Ci t
nx
m;nx
m¤n
upon separating
the diagonal terms
m D n and the off-diagonal terms m ¤ n.
ˇ
p ˇ
Since 2ˇak;x .m/ak;x .n/= mnˇ jak;x .m/j2 =m C jak;x .n/j2 =n, we see that the
off-diagonal terms above contribute
X jak;x .n/j2 X
1
jlog.m=n/j
n
k
k
nx
mx
m¤n
X jak;x .n/j2
X jak;x .n/j2
T
;
n
n
k
k
x k log.x k /
nx
nx
recalling that x k T = log T . The lemma follows upon noting that
X jak;x .n/j2
n
k
nx
X
D
X
p1 <<pr x ˛1 ;:::;˛r 1
P
˛i Dk
kŠ
X
X
2 ja.p /j2˛1 ja.p /j2˛r
k
1
r
˛1 ; : : : ; ˛r
p ˛1 pr˛r
1
p1 <<pr x ˛1 ;:::;˛r 1
P
˛i Dk
D kŠ
ja.p /j2˛1 ja.p /j2˛r
k
1
r
˛1 ; : : : ; ˛r
p ˛1 pr˛r
1
X ja.p/j2 k
:
p
px
In proving our theorem, we may assume that
p
10 log log T V 38 log T = log log T:
We also keep in mind that T is large. We define a parameter A by
81
log3 T
if V log log T;
ˆ
<2
log log T
AD
log3 T if log log T < V 21 .log log T / log3 T;
2V
ˆ
:
1
if V > 12 .log log T / log3 T:
MOMENTS OF THE RIEMANN ZETA FUNCTION
989
We further set x D T A=V and z D x 1= log log T . Using Lemma 2 and our proposition,
we find that
1C0
V C O.log log log T /;
logj. 12 C i t /j S1 .t / C S2 .t / C
2A
where
ˇX
ˇ
S1 .t / D ˇ
pz p
ˇ X
ˇ
S2 .t / D ˇ
1
ˇ
log.x=p/ ˇ
ˇ;
0
1
C log
Ci t log x
2
x
z<px
1
p
ˇ
log.x=p/ ˇ
ˇ:
0
1
C log
Ci t log x
2
x
If t 2 S.T; V / then we must either have
S2 .t / V
8A
S1 .t / V 1
or
7
DW V1 :
8A
By Lemma 3 we see that for any natural number k V =A 1, we have
Z 2T
X
k
1 k
T k.log log log T C O.1// :
jS2 .t /j2k dt T kŠ
p
T
z<px
Choosing k to be the largest integer below V =A 1, we obtain that the measure of
t 2 ŒT; 2T  with S2 .t / V =.8A/ is
2k
V
8A
(8)
T
.2k log log log T /k T exp
log V :
V
2A
Next we consider the measure of the set of t 2 ŒT; 2T  with S1 .t / V1 . By
Lemma 3, we see that for any natural number k log.T = log T /= log z,
Z 2T
X k
p k log log T k
1
jS1 .t /j2k dt T kŠ
T k
;
p
e
T
pz
so that the measure of t 2 ŒT; 2T  with S1 .t / V1 is
p k log log T k
T k
:
eV12
When V .log log T /2 , we choose k as the greatest integer less than V12 = log log T ,
and when V > .log log T /2 we choose k D Œ10V . It then follows that the measure
of t 2 ŒT; 2T  with S1 .t / V1 is
(9)
T p
V12 V
exp
C T exp. 4V log V /:
log log T
log log T
Our theorem follows upon combining the estimates (8) and (9).
990
KANNAN SOUNDARARAJAN
4. Moments of L-functions in families
We briefly sketch here the modifications needed to obtain bounds for moments
of L-functions in families. Throughout we assume the generalized Riemann hypothesis for the appropriate L-functions under consideration. If q is a large prime,
then Rudnick and Soundararajan [16] showed that, for positive rational numbers k,
X
2
jL. 12 ; /j2k k q.log q/k ;
.mod q/
where the sum is over primitive characters . The argument given here carries
2
over to obtain the upper bound k;" q.log q/k C" for all positive real k. The only
difference is that one uses the orthogonality relations of the characters .mod q/ to
treat nondiagonal terms in the analogue of Lemma 3.
More interesting is the case of quadratic Dirichlet L-functions. In [17] Rudnick
and Soundararajan showed that for rational numbers k 1
X[
L. 12 ; d /k k X.log X /k.kC1/=2 ;
jd jX
where the sum is over fundamental discriminants d , and d denotes the associated
primitive quadratic character. To obtain an upper bound, we seek to bound the
frequency of large values of L. 21 ; d /. Analogously to our proposition, we find
that1 , for any x 2 and with 0 as in our proposition,
X
ƒ.n/d .n/ log.x=n/ .1 C 0 / logjd j
1
:
log L. 12 ; d / C
C
O
0
1
log x
log x
2
log x
C log
x log n
2nx n 2
Notice that, in contrast to Lemma 2, the contribution of the prime squares in our
sum is 12 log log x, since d .p 2 / D 1 for all p − d . Taking this key difference
into account, we may argue as in Section 4, using now quadratic reciprocity and
the Pólya-Vinogradov inequality to develop the analogue of Lemma 3. Thus we
obtain that the number of d with jd j X and log L. 21 ; d / V C 12 log log X is
V2
X exp
.1 C o.1//
2 log log X
p
when log log X V D o..log log X / log3 X /; when V .log log X / log3 X this
number is X exp. cV log V / for some positive constant c. From these estimates
we deduce that
X[
L. 12 ; d /k k;" X.log X /k.kC1/=2C" :
jd jX
1 Since
we are assuming GRH we know that L. 12 ; d / 0. If L. 12 ; d / D 0 we interpret
as 1 so that the claimed inequality is still valid.
log L. 12 ; d /
991
MOMENTS OF THE RIEMANN ZETA FUNCTION
As a last example consider the family of quadratic twists of a given elliptic
P
s
curve E. We write the L-function for E as L.E; s/ D 1
nD1 a.n/n , where
the a.n/ are normalized so that ja.n/j d.n/ (the number of divisors of n). The
methods of [16] and [17] can be used to show that for rational numbers k 1,
X[
L.E ˝ d ; 21 /k k X.log X /k.k 1/=2 :
jd jX
Writing a.p/ D ˛p C ˇp with ˛p ˇp D 1 we obtain analogously to our proposition
that, for x 2 and 0 as before,
log L.E
˝ d ; 21 /
X
nDp ` x
`1
d .p ` /.˛p` C ˇp` / log.x=p ` /
`n1=2C0 =log x
log x
C .1 C 0 /
logjd j
1
:
CO
log x
log x
The contribution of the terms ` 3 above is O.1/, and the contribution of ` D 2
(the prime squares) is
X
p
p x
p −d
a.p 2 / 1 log.x=p 2 /
2p 1C20 =log x log x
1
2
log log x:
After taking this feature into account, we may develop the analogous argument
of Section 4. Thus, the number of d with jd j X and log L.E ˝ d ; 12 / V 21 log log X is
V2
.1 C o.1//
X exp
2 log log X
p
when log log X V D o..log log X / log3 X /; when V .log log X / log3 X this
number is X exp. cV log V / for some positive constant c. This leads to the
upper bound
X[
L.E ˝ d ; 12 /k X.log X /k.k 1/=2C" :
jd jX
Our work above is in keeping with conjectures of Keating and Snaith [10]
(see (51) and (79) there) that an analogue of Selberg’s result holds in families
of L-functions. Thus in the unitary family of .mod q/, we expect that the
distribution of logjL. 12 ; /j is Gaussian with mean 0 and variance 12 log log q.
In the symplectic family of quadratic Dirichlet characters, we expect that the
distribution of log L. 12 ; d / is Gaussian with mean 12 log logjd j and variance
log logjd j. Thus most values of L. 21 ; d / are quite large. In the orthogonal family
of quadratic twists of an elliptic curve, first we must restrict to those twists with
positive sign of the functional equation, or else the L-value is 0. In this restricted
992
KANNAN SOUNDARARAJAN
class, we expect that the distribution of log L.E ˝ d ; 21 / is Gaussian with mean
21 log logjd j and variance log log jd j. Thus the values in an orthogonal
family tend to be small. With a little more work, the ideas in this paper would
p show
(assuming GRH) that the frequency with which logjLj exceeds Mean C Var is
R1
2
bounded above by p1 e x =2 dx for any fixed real number . If in addition
2
to GRH, we assume that most of the L-functions under consideration do not have
a zero very near2 12 , then Selberg’s techniques would yield these Keating-Snaith
analogues.
Acknowledgments
I am grateful to Mike Rubinstein for some helpful remarks, and to Brian
Conrey for his encouragement.
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(Received December 5, 2006)
E-mail address: ksound@math.stanford.edu
S TANFORD U NIVERSITY , D EPARTMENT OF M ATHEMATICS , 450 S ERRA M ALL , B UILDING 380,
S TANFORD , CA 94305-2125, U NITED S TATES