Bendersky constants $A_k$

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Numbers

$k$

$A_k$, $\log A_k$

2.506628274631000502415765284811045253006986740609938316629923576342293654607841974946595838378057266

comment: $A_0=\sqrt{2\pi}$. $\log A_0=\frac12\log(2\pi)$.

0.9189385332046727417803297364056176398613974736377834128171515404827656959272603976947432986359541976

comment: $A_0=\sqrt{2\pi}$. $\log A_0=\frac12\log(2\pi)$.

1.030916752197392114193313096466942290633194306403487060227261741145166066978290405292931362554808852

0.03044845705839327078025153047115477664700048354497393625297188985903781794493689867779458488087449597

0.9795555269428446058242188372634918264455367524955299022577171427975885672481559614944444353833219619

-0.02065635413555207892219475198819162067344221752007328483722480100110227977570184736387288164860346104

0.9920479745250402600134369776254433567369048512761880893520946149155414538538945976180577361729564309

-0.007983811450268624280696670798789303905237693362298876417704739714028740281878657952543961969286982040

1.009680387285866161120089190462630692603276347211524918460924721562301425003410032770150565965276456

0.009633832541045196051551840709680435359814833852046082064381638441844295877911677818711960188946547067

1.005917196998673468444013983554255656390615655006932114009805157408146870342994632771967081708841469

0.005899759143515937450629877408392025579801534620157195865239392206359703759424905723023863007542258385

0.9899756533334170941753964830588692002082471514307453051285538624237746429596167574275668776364868292

-0.01007492874841218791896133807392106795952568370764601025279218527442883102590126151554818212974812205

0.9917183216328221969995474827657933398678597605730507924707659934095023793421761909309123888614061141

-0.008316161985602247359524426510534214225674122918829999040210532753056917407881235383483452514524403517

1.018469929920992912170659049376672172308610190564074920380070573675476194940638164855897954746833700

0.01830143236178910882275580939079223487501221181609616561277644447614296527625193418726816890048762231

10:

1.019110233329383853722164704986297513513481372840996044596414946765542895934843533415455512247816968

10:

0.01892992633814037422898050229034679523198525809516955581048623110070270515504148055235160734739912371

11:

0.9503312484532888665142338410153312715975664034561730408610888811622978491773444513652817966947442446

11:

-0.05094467258078564273401441728631221497701229143944943462571914754246873938405855250884696244801139056

12:

0.9386894455960125851529657813206767183332587685218350098663907163424058837380151170867640211773640195

12:

-0.06327058334146300059518230123430776751141818475323637667956594567062152546067497673747103437135461765

13:

1.222944251808133872647899960727717988561265803129532950108372810344606422768662030300126426921751143

13:

0.2012612725165693808602224647878112054974337909483877697611602603734091969446944591664369933309028831

14:

1.338644754241536299558046958873255142542092537062742480234062094589795315285196484552452931398735425

14:

0.2916577247438735203212240030702506669702630385330908321499093596565151870284637586775093924097273269

15:

0.3428308061328167365717111463406723781417269454832368772510761642419265535879711285213849602593289577

15:

-1.070518230097822017586384053620521361875761908404020549498798604972399628341823640964815558972026299

16:

0.1698183978427756077473095516831271187951529142863773586027175955000754217608888014719356708219630383

16:

-1.773025660899096396247787344189294481355419827646999177163917307737280926906655310456023712750519855

17:

1596.535085758038553851455236620441945331661100613504443414554639997110604534322956350654042110487568

17:

7.375590988586098900226655180156366003595132887402480114651943110394234973948799721572750181124798259

18:

929840.4118229400880431137850376043344453538159665584899793611480297248360368005905206171972143600780

18:

13.74276825021405443522056419051855107309537215770498560474565153488894633788585388234060990032339560

19:

1.782754276946714118817670977435445569560818370157206532448944355006293889658740667198406098411942863e-28

19:

-63.89422308883726052209728635290193473379204853571297702693070244186342305117484302240648214669460849

20:

3.557072551004370954018977147174911034638780085808891795757964400732114202429570120748913015783878382e-58

20:

-132.2809975042125145270982115857855186806480099995503145884745019241429157199404293877839464496253429

Definition

The Bendersky constants, also called generalized Glaisher constants, are the numbers $A_k$ [2], $k\geq0$, whose logarithms satisfy $\log A_k=\frac{B_{k+1}}{k+1}H_k-\zeta'(-k)$, where $H_k=\sum_{j=1}^k1/j$ and $H_0=0$. Each row gives the same constant in two conventions: first $A_k$, then $\log A_k$.

Parameters

$k$

— index ($k\geq0$)

Formulas

(1)

$\log A_k=\frac{B_{k+1}}{k+1}H_k-\zeta'(-k)$.

(2)

For $m\geq1$, $\log A_{2m}=(-1)^{m+1}\frac{(2m)!}{2(2\pi)^{2m}}\zeta(2m+1)$.

(3)

$\log A_0=\frac12\log(2\pi)$.

Comments

(4)

In the defining formula, the Bernoulli numbers are written as $B_n$.

(5)

For each listed index $k$, the two stored values are $A_k$ and $\log A_k$, in that order. They are one constant written in two conventions.

(6)

The asymptotic formula [2] for $\prod_{m=1}^n m^{m^k}$ has $A_k$ as its constant factor: $A_0$ is the constant in Stirling's formula for $n!$, and higher $A_k$ normalise the corresponding higher products. Dowker [1] uses the name Glaisher-Kinkelin-Bendersky constants.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.arith.misc import bernoulli
from sage.rings.complex_arb import ComplexBallField
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ

CBF = ComplexBallField(numberdb.bits(100, losing=96))
k = ZZ(2)
H = sum(QQ(1) / QQ(j) for j in range(1, int(k) + 1))
log_A = CBF(QQ(bernoulli(k + 1)) * H / QQ(k + 1)) - CBF(-k).zetaderiv(1)
log_A.exp(), log_A

References

[1]

J. S. Dowker, Computation of the derivative of the Hurwitz zeta-function and the higher Kinkelin constants, 2015. (arXiv)

Links

[2]

Wikipedia: Glaisher-Kinkelin constant

[3]

MathWorld: Glaisher-Kinkelin Constant

Similar tables

Glaisher-Kinkelin constant $A$ — the classical Glaisher-Kinkelin constant is the $k=1$ member $A_1$ of this family and is held separately under its own name, together with its logarithmic convention; a reader with the name $A$ or a value of $A_1$ usually wants that table

Values of the Riemann zeta function at rational numbers — the even-index constants are given by zeta values at positive odd integers

Bernoulli numbers — the defining formula uses the Bernoulli numbers $B_{k+1}$

Factorial of natural numbers — $A_0=\sqrt{2\pi}$ is the constant in Stirling's formula $n!\sim\sqrt{2\pi n}\,(n/e)^n$

Data properties

Entries are of type: real number

How well the digits are known: proven

How they were obtained:

Each value is computed as a Sage complex ball with arb at numberdb.bits(digits, losing=96) bits. The imaginary part is required to contain zero before the real part is returned.

The entries are checked against (2), Wikipedia's tabulated decimal values for $0\leq k\leq10$, and the Barnes $G$-function identity in the companion table of the Glaisher-Kinkelin constant.

Table is complete: no (it holds $A_k$ and $\log A_k$ for $k=0$ and every integer $2\leq k\leq20$; the classical member $A_1$ is in the table of the Glaisher-Kinkelin constant $A$)