Uniform asymptotics for the full moment conjecture of the Riemann zeta function

0.67 MB·English

by  Ghaith A. Hiary

Uniform asymptotics for the full moment
conjecture of the Riemann zeta function
2
Ghaith A. Hiary and Michael O. Rubinstein ∗
1
0
2

n
a Abstract
J
Conrey,Farmer,Keating,Rubinstein,andSnaith,recentlyconjectured
4
formulas for the full asymptotics of the moments of L-functions. In the

case of the Riemann zeta function, their conjecture states that the 2k-th
]
T absolute moment of zeta on the critical line is asymptotically given by a
N certain2k-foldresidueintegral. Thisresidueintegralcanbeexpressedas
a polynomial of degree k2, whose coefficients are given in exact form by
.
h elaborate and complicated formulas.
t In this article, uniform asymptotics for roughly the first k coefficients
a
m of the moment polynomial are derived. Numerical data to support our
asymptoticformulaarepresented. Anapplicationtoboundingthemaxi-
[
mal size of the zeta function is considered.

3
v
Contents
2
5
3 1 Introduction 2
4
1.1 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
.
6 1.2 Numericalverificationsandanapplicationtothemaximalsizeof
0 |ζ(1/2+it)|. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1
1
2 Proof of the main theorem 11
:
v
i 3 An algorithm to reduce to the first half 17
X
3.1 The first step: from p (α) to p (λ;0) . . . . . . . . . . . . . . . . 18
k k
r
a 3.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 The second step: from p (λ;0) to N0(λ) . . . . . . . . . . . . . . 20
k k
4 An algorithm to compute N0(λ) 21
k
4.1 An algorithm to compute N0(λ) . . . . . . . . . . . . . . . . . . 25
k
4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Applications of the algorithms 27
∗Both authors are supported by the National Science Foundation under awards DMS-
0757627 (FRG grant) and DMS-0635607. In addition, the second author is supported by an
NSERCDiscoveryGrant.
1
6 The arithmetic factor 31
6.1 Contribution of “the small primes”: via Cauchy’s estimate . . . . 33
6.1.1 The combinatorial sum . . . . . . . . . . . . . . . . . . . 33
6.1.2 The convergence factor sum . . . . . . . . . . . . . . . . . 43
6.2 Contribution of “the large primes”: via Taylor expansions . . . . 44
6.2.1 The combinatorial sum . . . . . . . . . . . . . . . . . . . 44
6.2.2 The convergence factor sum . . . . . . . . . . . . . . . . . 48
6.3 Bounding the coefficients of the arithmetic factor . . . . . . . . . 49
7 The product of zetas 51
1 Introduction
The absolute moments of the Riemann zeta function on the critical line are
a natural statistical quantity to study in connection with value distribution
questions. For example, they can be used to understand the maximal size of
the zeta function. These moments are also connected to the remainder term in
the general divisor problem [T].
Hardy and Littlewood proved a leading-term asymptotic for the second mo-
ment on the critical line [HL]. A few years later, in 1926, Ingham gave the
full asymptotic expansion [I]. In the same article, Ingham gave a leading term
asymptoticforthefourthmoment. Thefullasymptoticexpansionforthefourth
momentwasobtainedbyHeath-Brownin1979[HB]. Incomparison,thehigher
moments seemed far more difficult and mysterious. Keating and Snaith, in a
breakthrough, conjectured the leading-term asymptotic [KS].
Recently,however,basedonnumber-theoreticconsiderations,Conrey,Farmer,
Keating, Rubinstein, and Snaith, conjectured [CFKRS1] [CFKRS2] the follow-
ing full asymptotic expansion for the 2k-th absolute moment of the Riemann
zeta function ζ(s) on the critical line:
1 (cid:90) T 1 (cid:90) T (cid:18) t (cid:19)
|ζ(1/2+it)|2kdt∼ P log dt, as T →∞, (1)
T T k 2π
0 0
where P (x) is a polynomial of degree k2:
k
P (x)=:c (k)xk2 +c (k)xk2−1+···+c (k), (2)
k 0 1 k2
given implicitly by the 2k-fold residue
(−1)k 1 (cid:73) (cid:73) G(z ,...,z )∆2(z ,...,z )
P (x)= ··· 1 2k 1 2k (3)
k k!2 (2πi)2k (cid:81)2k z2k
i=1 i
×ex2(cid:80)ki=1zi−zk+idz1...dz2k,
where the path of integration is around small circles enclosing z =0, and
i
(cid:89)
∆(z ,...,z ):= (z −z ) (4)
1 2k j i
1≤i<j≤2k
2
is the Vandermonde determinant, and
k
(cid:89)
G(z ,...,z ):=A(z ,...,z ) ζ(1+z −z ), (5)
1 2k 1 2k i k+j
i,j=1
is a product of zetas and the “arithmetic factor” (Euler product)
A(z ,...,z )
1 2k
:=(cid:89) (cid:89)k (cid:0)1−p−1−zi+zk+j(cid:1) (cid:90) 1(cid:89)k (cid:18)1− e2πiθ (cid:19)−1(cid:18)1− e−2πiθ (cid:19)−1dθ
p i,j=1 0 j=1 p21+zj p21−zk+j
(6)
k
(cid:89)
(1−p−1+zi+k−zm)
k
=(cid:89)(cid:88)(cid:89)m=1 . (7)
1−pzi+k−zj+k
p j=1i(cid:54)=j
As pointed out by [CFKRS1], the rhs of (3) has an almost identical form to
an exact expression for the moment polynomial of random unitary matrices,
the difference being that G(z ,...,z ) is replaced by the function (cid:81)k (1−
1 2k i,j=1
ezj+k−zi)−1 in the unitary case, so there is no arithmetic factor.
The CFKRS conjecture (3) agrees with the theorems of Hardy and Little-
wood, Ingham, and Heath-Brown, for k = 1 and k = 2. It has been supported
numerically; see [CFKRS1], [CFKRS2] [HO] [RY]. The conjecture provides a
method for computing the lower order coefficients of the moment polynomial
P (x). It gives, in particular, a stronger asymptotic than that of Keating and
k
Snaithwho, bycarryingoutananalogouscomputationforrandomunitaryma-
trices, first predicted the leading coefficient (see [KS]):
a g
c (k)= k k , (8)
0 k2!
where
a :=(cid:89)(1−1/p)k2F(k,k;1;1/p), (9)
k
p
and
k−1
(cid:89) j!
g :=k2! . (10)
k (j+k)!
j=0
1.1 Results
Our main theorem develops a uniform asymptotic for c (k) in the region 0 ≤
r
r ≤ kβ, for any fixed β < 1. We expect the asymptotics can be corrected so
as to remain valid well beyond the first k coefficients (i.e. for β ≥1), and that
the methods in our paper, which are of combinatorial nature, will be helpful in
deriving uniform asymptotics for the moments of other L-functions.
3
To state our main theorem, let us first define
(cid:88)klogp F(k+1,k+1;2;1/p) logp
B := − , (11)
k p−1 F(k,k;1;1/p) p
p
where F(a,b;c;t) is the Gauss hypergeometric function
Γ(c) (cid:88)∞ Γ(a+n)Γ(b+n) tn
F(a,b;c;t):= . (12)
Γ(a)Γ(b) Γ(c+n) n!
n=0
In the notation of [CFKRS2], B is the same as B (1;), which is given in
k k
Eqs. (2.24) and (2.43) there. The factor B is arithmetic in nature. It is the
k
coefficientofthelinearterminthefollowingTaylorexpansionofthearithmetic
factor:
k
(cid:88)
logA(z ,...,z )=loga +B z −z +··· , (13)
1 2k k k i k+i
i=1
where it is known (see 2.7 of [CFKRS1]) that
a =A (0,...,0). (14)
k k
Theorem 6.2 will later furnish the following asymptotic for B :
k
B ∼2k logk, as k →∞. (15)
k
Main theorem. Fix β <1, let 0≤r ≤kβ, and let
τ :=2B +2γk, (16)
k k
where γ =0.5772... is the Euler constant. Notice by (15) we have
τ ∼4k logk, as k →∞. (17)
k
Then as k →∞, and uniformly in 0≤r ≤kβ,
(cid:18)k2(cid:19)a g (cid:20) (cid:18)r2(cid:19)(cid:21)
c (k) = τr k k 1+O (18)
r k r k2! k2
(cid:18)k2(cid:19) (cid:104) (cid:16) (cid:17)(cid:105)
= τr c (k) 1+O k2(β−1) . (19)
k r 0
Alternatively,
τrk2r (cid:104) (cid:16) (cid:17)(cid:105)
c (k)= k c (k) 1+O k2(β−1) , (20)
r r! 0
as k →∞. Asymptotic constants depend only on β.
4
Remarks: 1)Theasymptoticformulas(18)and(19)ofourtheoremareactually
equalities for r = 0, and r = 1. The r = 0 case is trivial, and the r = 1 case
follows from either (2.71) of [CFKRS2] or (49) below. 2) For comparison, the
corresponding asymptotic in the unitary case, provided in [HR], is:
(cid:18)k2(cid:19) (cid:20) (cid:18)r2(cid:19)(cid:21)
c˜ (k)=kr c˜ (k) 1+O , (21)
r r 0 k2
wherec˜ (k)isthecoefficientofxk2−r inthe2k-thmomentpolynomialofrandom
r
unitary matrices.
Although the CFKRS conjecture seems hopelessly difficult to prove, the
precise nature of the asymptotic formula allows one to gain insight into the
behaviorofthezetafunction. Forexample, byderivinganasymptoticforc (k)
r
that is applicable as r and k both tend to infinity, one can understand the true
size of ζ(1/2+it). The results we present here are a step in this direction.
OnedifficultyinextractinguniformasymptoticsforthecoefficientsofP (x)
k
from a residue like (3) is that the coefficients are given only implicitly. By
comparison, both the coefficients and the roots of the moment polynomials
for random unitary matrices, which correspond to the zeta-function moment
polynomials according to the random matrix philosophy, are known explicitly,
viarandommatrixtheorycalculations. Infact,theproofofTheorem1of[HR],
which provides complete uniform asymptotics for the coefficients in the unitary
case, makes essential use of the information about the roots via a saddle-point
technique. In the case of the zeta function, however, we do not have ‘simple’
closed form expressions for the moment polynomials.
We remark that if one directly applies the methods of this paper to the
residue expression for unitary moment polynomials, given in [CFKRS1] Eq.
(1.5.9), then one encounters similar difficulties as in the zeta function (e.g. a
similar difficulty in deriving asymptotics beyond the first k coefficients). The
mainaddedsimplicityintheunitarycaseisthatitdoesnotinvolveanarithmetic
factor.
Before delving into the careful details of the next sections, let us describe
the basic idea of the proof. To this end, define
k
(cid:89)
R(z ,...,z ):=G(z ,...,z ) (z −z ). (22)
1 2k 1 2k i k+j
i,j=1
where, recall, G(z ,...,z )=A(z ,...,z )(cid:81)k ζ(1+z −z ). The extra
1 2k 1 2k i,j=1 i k+j
product on the rhs in (22) is introduced in order to cancel the poles in the
product of zetas in the definition of G(z ,...,z ). This renders the function
1 2k
R(z ,...,z ) analytic and non-zero in a neighborhood of the origin, where it
1 2k
5
is equal to a . Therefore, we may write
k
P (x)= (−1)k 1 (cid:73) ···(cid:73) ∆2(z1,...,z2k)ex2(cid:80)ki=1zi−zk+i
k k!2 (2πi)2k (cid:81)k (z −z )(cid:81)2k z2k
i,j=1 i k+j i=1 i
(23)
×elogR(z1,...,z2k)dz1...dz2k,
and consider the Taylor expansion of logR(z ,...,z ):
1 2k
k
logR(z ,...,z )=loga + τk (cid:88)z −z +··· , (24)
1 2k k 2 i k+i
i=1
where, recall, τ =2B +2γk ∼4klogk, as k →∞. Also, dropping the factor
k k
exp(logR(z ,...,z )), define
1 2k
p (x,0):= (−1)k 1 (cid:73) ···(cid:73) ∆2(z1,...,z2k)ex2(cid:80)ki=1zi−zk+i dz ...dz (25)
k k!2 (2πi)2k (cid:81)k (z −z )(cid:81)2k z2k 1 2k
i,j=1 i k+j i=1 i
(a more general function p (x,α) will be introduced in the next section). Our
k
basic claim is that the approximation
P (x)≈a p (x+τ ,0), (26)
k k k k
obtained from P (x) by truncating the Taylor expansion of logR(z ,...,z )
k 1 2k
at the linear term, is good enough to deduce asymptotics for the coefficients
{c (k), 0≤r ≤kβ}, for any fixed β <1, in the sense the leading term asymp-
r
totic of the coefficient of xk2−r, 0≤r ≤kβ, on either side of (26) is the same.
Notice the formula defining p (x,0) does not involve the complicated arith-
k
metic factor A(z ,...,z ) present in the residue expression for P (x). More-
1 2k k
over, by the results of Conrey, Farmer, Keating, Rubinstein, and Snaith, the
function p (x+τ ,0) can be evaluated explicitly as a polynomial in x of degree
k k
k2. For, by property (45) later, and the formulas in §2.7 of [CFKRS1], we have
g
p (x+τ ,0)= k (x+τ )k2. (27)
k k k2! k
The idea that the linear term in the Taylor expansion of logR(z ,...,z )
1 2k
ought to dominate over 0 ≤ r ≤ kβ was inspired, in part, by the analogous
asymptotic (21), derived in [HR], for the moments of the characteristic polyno-
mial of random unitary matrices.
As mentioned earlier, the main theorem of this paper shows that the coef-
ficients of the polynomial a p (x+τ ,0) = akgk(x+τ )k2 provide the leading
k k k k2! k
asymptotics, as k → ∞, for essentially the first k coefficients of P (x). The
k
proof of this theorem will naturally split into two main parts. In the first part,
whichispresentedin§3, §4, and§5, weobtainestimatesoncertainfunctionsin
k, later denoted by p . In the second part, which is presented in §6, we obtain
k
6
boundsontheTaylorcoefficientsofthelogarithmofthearithmeticfactor. The
latter bounds (and in some cases asymptotics) are fairly involved but generally
straightforward, while the former bounds are more subtle, requiring somewhat
more thought. Both bounds are obtained via essentially combinatorial argu-
ments.
1.2 Numerical verifications and an application to the max-
imal size of |ζ(1/2+it)|.
Table 1 provides numerical confirmation of our Main Theorem, listing values of
the ratio
c (k)
r (28)
c (k)(cid:0)k2(cid:1)τr
0 r k
for k = 10,20,30,40,50 and 0 ≤ r ≤ 7. Our theorem provides an estimate for
this ratio of 1+O(cid:0)(r/k)2(cid:1), and our table is consistent with such a remainder
term, agreeing, for example, to 3-4 decimal places for r = 2 and k = 50, and
2-3 decimal places for r =8 and k =50.
Next, let β < 1, and, as usual, k ∈ Z . While the asymptotic formula
≥0
for c (k) given in our Main Theorem holds, as k → ∞, for r < kβ, it appears,
r
numerically, that our asymptotic formula is, uniformly, an upper bound for
|c (k)| for all 0≤r ≤k2.
r
We therefore conjecture, for all non-negative integers k, and all 0≤r ≤k2,
that:
(cid:18)k2(cid:19)
|c (k)|≤c (k) τr. (29)
r 0 r k
We have verified this conjecture numerically for all k ≤13, 0≤r ≤k2, and all
k ≤64, 0≤r ≤8. The coefficients of the moment polynomials were computed
in the former case in [RY] and in the latter case using the program developed
for the computations in [CFKRS1] and [CFKRS2]. See Figure 1 for evidence
supportingthisconjecture,whichdepictstheratioc (k)/(cid:16)c (k)(cid:0)k2(cid:1)τr(cid:17)fork =
r 0 r k
10 and 0≤r ≤k2.
Assuming the bound (29), we have, by the binomial theorem and term-wise
comparison, the following upper bound for P (x), for all k ∈Z and x∈R:
k ≥0
|P (x)|≤c (k)(|x|+τ )k2. (30)
k 0 k
Let |ζ(1/2+it )| = m := max |ζ(1/2+it)|. Lemma 3.3 of [FGH]
0 T t∈[0,T]
provides:
(cid:32) (cid:33)1/2k
1 (cid:90) T
m ≤2(CT logT)1/2k |ζ(1/2+it)|2kdt (31)
T T
0
for some absolute constant C > 0. Farmer, Gonek and Hughes use this in-
equality, combinedwiththeKeatingandSnaithleadingtermconjectureforthe
7
k r cr(k) cr(k)/(cid:16)c0(k)(cid:0)kr2(cid:1)τkr(cid:17)
10 0 3.548884925e-148 1
10 1 2.357691331e-144 1
10 2 7.702336630e-141 0.9934255388
10 3 1.649486344e-137 0.9803060865
10 4 2.604519447e-134 0.9608017974
10 5 3.233666778e-131 0.9352015310
10 6 3.287651416e-128 0.9039165203
10 7 2.814729470e-125 0.8674698258
20 0 9.404052083e-789 1
20 1 7.007560591e-784 1
20 2 2.600909647e-779 0.9986738069
20 3 6.410977573e-775 0.9960221340
20 4 1.180624032e-770 0.9920509816
20 5 1.732651855e-766 0.9867716274
20 6 2.110801042e-762 0.9802005819
20 7 2.195579847e-758 0.9723595087
30 0 2.174528185e-2019 1
30 1 6.409313254e-2014 1
30 2 9.429995281e-2009 0.9994621075
30 3 9.234275546e-2004 0.9983864033
30 4 6.770756592e-1999 0.9967738368
30 5 3.964993050e-1994 0.9946262257
30 6 1.931729883e-1989 0.9919462534
30 7 8.053463103e-1985 0.9887374636
40 0 1.878520688e-3887 1
40 1 1.450126078e-3881 1
40 2 5.592030026e-3876 0.9997132915
40 3 1.436301603e-3870 0.9991398909
40 4 2.764308226e-3865 0.9982800615
40 5 4.252265871e-3860 0.9971343131
40 6 5.445979160e-3855 0.9957034019
40 7 5.972928889e-3850 0.9939883295
50 0 3.461963190e-6425 1
50 1 5.605367518e-6419 1
50 2 4.535291006e-6413 0.9998231027
50 3 2.444917857e-6407 0.9994693125
50 4 9.879474579e-6402 0.9989387280
50 5 3.191850197e-6396 0.9982315414
50 6 8.588531004e-6391 0.9973480389
50 7 1.979690769e-6385 0.9962886003
Table 1: A comparison of our asymptotic formula for cr(k), for k = 10,20,30,40,50 and
r≤7. The1’sareexplainedbytheremarkfollowingtheMainTheoremthattheasymptotic
formulaisactuallyanidentityforr=0andr=1. Weexpecttheretobelowertermsinour
asymptoticexpansion,andwillreturntotheproblemofdeterminingtheminafuturepaper.
8
1.2
1
0.8
c
oti 0.6
pt
m
y
s
a
_r(k)/ 0.4
c
0.2
0
-0.2
0 20 40 60 80 100
r
Figure 1: We compare the ratio of cr(10), 0 < r < 100, to our asymptotic formula. Here,
k=10isrelativelysmall,andweonlygetreasonableagreementforthefirstfewr. However,
thegraphindicatesthattheasymptoticformulais,uniformly,anupperboundfor|cr(k)|.
9
moments of zeta to estimate m . However, the leading term does a poor job at
T
bounding the true size of the moments if we allow k to grow with T.
However, using our conjectured bound (30) for P (x), we have, in whatever
k
range of k that (1) remains valid asymptotically, that
(cid:32) (cid:33)1/2k
1 (cid:90) T
m ≤2(c (k)C T logT)1/2k (|log(t/2π)|+τ )k2dt (32)
T 0 2 T k
0
for some absolute constant C > 0. Following the argument in [FGH], we will,
2
at the end, apply the above with k proportionate to (log(T)/loglog(T))1/2.
The portion of the integral, t ∈ (0,2π) where log(t/2π) is negative con-
(cid:16) (cid:17)
tributes O (k2)k2 , on using: (cid:82)2π|log(t/2π)|k2dt = 2πk2!, the binomial ex-
0
pansion, Stirling’s formula for k2!, and also (cid:80)k2τr/r! < exp(τ ) combined
0 k k
with (17). (We could also slightly modify the argument in [FGH] and ignore
this interval outright.)
Next, by (17), we have τ = O(klogk). Thus, if k ≤ C log(T)/loglog(T),
k 3
for some absolute constant C , the contribution to the integral for t∈[2π,T] is
3
(cid:16) (cid:17)
O T(C log(T))k2 , for some absolute constant C .
4 4
Therefore, if k = O(log(T)1/2), we can ignore the portion of the integral
from 0 to 2π, and get:
m (cid:28)2(c (k)C T logT)1/2k(C logT)k/2 (33)
T 0 5 4
for some absolute constant C , i.e.
5
logc (k) log(T)+loglog(T) k
logm (cid:28) 0 + + loglogT +O(k). (34)
T 2k 2k 2
Combining Conrey and Gonek’s estimate [CG]:
loga ∼−k2log(2eγlogk)+o(k2) for k →∞, (35)
k
with the asymptotics of the Barnes G-function, see (3.17) and (3.18) of [FGH],
gives:
logc (k) klogk
0 =− +O(kloglogk). (36)
2k 2
Hence,
log(T)+loglog(T) k klogk
logm (cid:28) + loglogT − +O(kloglogk), (37)
T 2k 2 2
i.e. bound(3.20)of[FGH]continuestoholdevenwhenweuseourupperbound
for the moment polynomials, rather than the much smaller and less precise (as
k grows) leading term.
Taking, as in [FGH], k ∼c(log(T)/loglog(T))1/2, and choosing the optimal
c=21/2, thus gives the identical upper bound (3.9) of [FGH]:
(cid:18) (cid:18)(logT)1/2logloglogT(cid:19)(cid:19)
m (cid:28)exp (1logT loglogT)1/2+O . (38)
T 2 (loglogT)1/2
10