Glaisher-Kinkelin constant $A$

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Numbers

convention

$A$, $\log A$

$A$:

1.282427129100622636875342568869791727767688927325001192063740021740406308858826461129736491958202374

comment: The classical Glaisher-Kinkelin constant.

$\log A$:

0.2487544770337842625472529935761139760973697136685351169998556396906930329999105060928584336658420889

comment: $\log A_1=\frac1{12}-\zeta'(-1)$.

Definition

The Glaisher-Kinkelin constant is the classical constant $A=A_1$ [3] in the Bendersky family. The two entries give the same number in two conventions: $A$ and $\log A$.

Parameters

convention

— $A$ or $\log A$

Formulas

(1)

$\log A=\frac1{12}-\zeta'(-1)$.

(2)

$G(1/2)$ satisfies $G(1/2)=2^{1/24}e^{1/8}A^{-3/2}\pi^{-1/4}$.

Comments

(3)

$A$ and $\log A$ are stored together because they are one constant in two conventions.

(4)

The constant $A$ is the normalising constant for the hyperfactorial product $\prod_{m=1}^n m^m$.

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(1)
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

Bendersky constants $A_k$ — $A$ is the $k=1$ member $A_1$ of the Bendersky family; the family table holds the other displayed indices, also with each constant beside its logarithmic convention, and is the table to use for a general indexed $A_k$

Values of the Barnes $G$-function at rational numbers — $A$ appears in closed forms for special values of $G$, including $G(1/2)$

Factorial of natural numbers — $A$ is the hyperfactorial analogue of the Stirling constant $\sqrt{2\pi}$ for $n!$

Data properties

Entries are of type: real number

How well the digits are known: proven

How they were obtained:

Both values are computed as Sage complex balls with arb at numberdb.bits(digits, losing=96) bits. The imaginary part is required to contain zero before the real part is returned, and $A$ is checked against (2).

Table is complete: yes