Leading constants $c_k$ of the moments of the Riemann zeta function

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Numbers

$k$

$c_k$

1/2:

1.129928745332153348965077876235947848618728488855121731617025575008328464965697827571343481273840529

comment: The second moment has $a_1=f_U(1)=1$, so $c_1=1$.

3/2:

0.4123536679132824095756949439043794109561795313936527962118652346669921040740369613184494185803962262

0.05066059182116888572193973160486381945217938733612327442280451594708656048111772059560463696281091881

comment: The fourth moment has $a_2=6/\pi^2$ and $f_U(2)=1/12$, so $c_2=1/(2\pi^2)$.

5/2:

0.001319454862949434371589967708367067841991748955830149852674101634335953117729556691682573226058192226

0.000005708527034652788398376841445252313559397328900532833950983032039199246535871335734526769079053203822

7/2:

3.422023855644058232390576710128913209778745110803149139024296386050839847066396684159658456654578147e-9

2.465018391934227354079893840260574929108176166341223638407021065777411150520576672162196414146774585e-13

9/2:

1.898458090468574223660900490279813200326353717060276604800649927896403450864455621604203397367715414e-18

1.416001020622731200955087648458784870984754017274309738448961729206700072070266047231892533105885314e-24

11/2:

9.387938398962274838608294704492705414880977194592905386249468958576717931759628926249271660376891159e-32

5.129473409149191124314650595531376614404313832458118736959272123400766383196476287049853173336522532e-40

6.582284787600549937846094301236278279116673247347136373382467961534789490488447715555850786082214677e-60

1.870442160116884420247132953797897812426294969595083854867598414829013235739080527259203913344649014e-84

7.920155238368529031678692249580806272909597679329446294550691035407972599765784320916005531889436926e-114

10:

3.548884924773034809924700340454319530928856545802717937219576986105420387197551039589800144687801910e-148

11:

1.245131388165943091406248531056891132685096123251528549037600957789806879460395453761361790338241028e-187

12:

2.614375653006404194243944641431186993702075126009896391700116028815673631228888194293940588085024262e-232

Definition

For $k>0$ rational, $c_k=a_kG(1+k)^2/G(1+2k)$, where $a_k$ is in (3) and $G$ is the Barnes $G$-function. It is the leading constant in the averaged $2k$-th moment asymptotic for $|\zeta(1/2+it)|$ (1).

Parameters

$k$

— half moment exponent ($k>0$)

Formulas

(1)

Conjecturally, $\frac1T\int_0^T |\zeta(1/2+it)|^{2k}\,dt\sim c_k(\log T)^{k^2}$.

(2)

$c_k=a_k f_U(k)$, where the random-matrix factor is $f_U(k)=G(1+k)^2/G(1+2k)$.

(3)

$a_k=\prod_p \left(1-\frac1p\right)^{k^2} \sum_{m=0}^{\infty} \left(\frac{\Gamma(m+k)}{m!\Gamma(k)}\right)^2p^{-m}$, where the product runs over rational primes [2].

(4)

If $k$ is a positive integer, then $a_k=\prod_p \left(1-\frac1p\right)^{(k-1)^2}\sum_{j=0}^{k-1}{k-1\choose j}^2p^{-j}$ [2].

(5)

If $k$ is a positive integer, then $f_U(k)=\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}$ and $(k^2)!f_U(k)=1,2,42,24024,701149020,\ldots$ for $k=1,2,3,4,5,\ldots$.

(6)

$c_1=1$ and $c_2=1/(2\pi^2)$.

Comments

(7)

The parameter is $k$, not the full exponent $2k$: $k=1$ is the second moment and $k=2$ is the fourth moment.

(8)

Here $\zeta$ is the Riemann zeta function [3]. The product defining $a_k$ and the Barnes $G$-function factor define $c_k$ unconditionally. The asymptotic in (1) is a theorem for $k=1$ and $k=2$; every other $k$ here is a constant in the Keating-Snaith and CFKRS conjectures [1] [2].

Programs

(P1)

PARI/GP

\\ positive integer k only: the finite Euler-product formula is not valid for half-integers
default(realprecision, 140)
k = 3
a = prodeulerrat((1 - 1/p)^((k - 1)^2) * \
  sum(j = 0, k - 1, binomial(k - 1, j)^2/p^j), 1, 2)
f = prod(j = 0, k - 1, factorial(j)/factorial(j + k))
a * f

References

[1]

J. P. Keating and N. C. Snaith, Random matrix theory and $\zeta(1/2+it)$, Communications in Mathematical Physics 214 (2000), 57-89. (doi)

[2]

J. Brian Conrey, David W. Farmer, Jon P. Keating, Michael O. Rubinstein and Nina C. Snaith, Integral moments of L-functions, Proceedings of the London Mathematical Society 91 (2005), 33-104. (arXiv)

Links

[3]

Wikipedia: Riemann zeta function

[4]

PARI/GP reference: prodeulerrat

Similar tables

Arithmetic factors $a_k$ in the moments of the Riemann zeta function — stores the Euler-product factor $a_k$ in the same normalisation

Values of the Barnes $G$-function at rational numbers — contains the Barnes $G$-function values used in $f_U(k)=G(1+k)^2/G(1+2k)$

Values of the Riemann zeta function at rational numbers — contains $\zeta(2)$, which gives the control value $c_2=1/(2\pi^2)$

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds $k=1/2,1,3/2,\ldots,6$ and every integer $7\leq k\leq12$)

How they were obtained:

The generator computes $a_k$ by the same Euler-product method as the arithmetic-factor table, multiplies by the Barnes $G$-function factor computed in arb ball arithmetic, and writes only digits that agree between 140-digit and 160-digit computations.

The checks include $c_1=1$, $c_2=1/(2\pi^2)$, the integer rows against PARI's prodeulerrat [4], and the computed $a_k$ values against the arithmetic-factor table.