Arithmetic factors $a_k$ in the moments of the Riemann zeta function

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Numbers

$k$

$a_k$

1/2:

0.9883590825750497440110552755486309770794952184053771383224622488566275158970094853496287445952250972

comment: Every local factor is $1$, so $a_1=1$.

3/2:

0.9184884018895747701784434711017771309947266423371875086164791347528223064367420443402602205751523399

0.6079271018540266286632767792583658334261526480334792930736541913650387257734126471472556435537310257

comment: At $k=2$ the local factor collapses to $1-p^{-2}$, so $a_2=1/\zeta(2)=6/\pi^2$.
equals: $6/\pi^2$

5/2:

0.2394905853018649310394770921993686654897564396817820491554980459464151284053435978386878244299854532

0.04932167357940009176197591008697998915319292170060368533649339681868149006992834074631128484301968102

7/2:

0.004858903408918558515640590398972113568214621303571182739091014955639638719989094928055815563136286688

0.0002146814097756221813396428504209017832659459921380567761372335530438336699922176470706122803437395747

9/2:

0.000004048729868663666419277617982687685460283295249491143205355393251013626344514315617070586849339473583

3.132556507381303165695050918567179739839977642213216037630307010012639654335529938593124145590841259e-8

11/2:

9.622779874212624100220853097149492785670824801471484392243819463105515978170487236155710451054836624e-11

1.141469703544542068559601046893471950034154897462422406579934523590203830246702225682425384131852025e-13

8.427923062936502051118306894141000693559285755068639304721729793240334460899783542860594070837923217e-21

1.074818337204139589551572399558077537482774448977130100905855077879394234073180017802033015399667487e-29

2.083394629387804465402773428952723823396248519384845811710452174961200637949227401953894941638512600e-40

10:

5.521274498002299073256688160551070939410381566452275725209430558965284558408125848241678771060548339e-53

11:

1.828542529836930562516819009002879807746638851524509341816015743869149644555248020649524176146900370e-67

12:

7.001763863028555681200875931763587301479238075549162359436456743581256718672216111117550111681198129e-84

Definition

For $k>0$ rational, $a_k$ is the Euler product in (2). It is the arithmetic factor in (1) for the averaged $2k$-th moment of $\zeta(s)$ [2].

Parameters

$k$

— half moment exponent ($k>0$)

Formulas

(1)

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

(2)

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

(3)

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

(4)

$a_1=1$, and $a_2=\prod_p(1-p^{-2})=1/\zeta(2)=6/\pi^2$.

Comments

(5)

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

(6)

The leading constant in (1) factors as $c_k=a_k f_U(k)$, where $f_U(k)=G(1+k)^2/G(1+2k)$ is the random-matrix factor and $G$ is the Barnes $G$-function. This table stores $a_k$ alone.

(7)

The asymptotic in (1) is known for $k=1$ and $k=2$ and is conjectural for the other entries [1]. The arithmetic factor $a_k$ is defined by (2) for every $k>0$.

Programs

(P1)

PARI/GP

\\ positive integer k only: this finite product is not valid for half-integers
default(realprecision, 140)
k = 3
prodeulerrat((1 - 1/p)^((k - 1)^2) *
  sum(j = 0, k - 1, binomial(k - 1, j)^2/p^j), 1, 2)

References

[1]

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

[2]

Wikipedia: Riemann zeta function

[3]

PARI/GP reference: prodeulerrat

Similar tables

Named rational Euler products over primes — contains $a_2=6/\pi^2$ as the squarefree-density product in another context

Values of the Riemann zeta function at rational numbers — contains $\zeta(2)$, which gives the control value $a_2=1/\zeta(2)$

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

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 the small-prime local factors directly with arb's hypergeometric function, computes the middle prime range from the exact power-series coefficients of the logarithm of the local factor in (2), and computes the remaining prime tail from prime-zeta values.

The written 100 digits were computed with 140-digit settings and recomputed with 160-digit settings; the written digits agreed. The checks include $a_1=1$, $a_2=6/\pi^2$, and the integer rows against PARI's prodeulerrat [3] applied to (3).