Glaisher–Kinkelin constant

In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to special functions like the K-function and the Barnes G-function. The constant also appears in a number of sums and integrals, especially those involving the gamma function and the Riemann zeta function. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin.

Its approximate value is:

A = 1.28242712910062263687...   (sequence A074962 in the OEIS).

Glaisher's constant plays a role both in mathematics and in physics. It appears when giving a closed form expression for Porter's constant, when estimating the efficiency of the Euclidean algorithm. It also is connected to solutions of Painlevé differential equations and the Gaudin model.^[1]

The Glaisher–Kinkelin constant A can be defined via the following limit:^[2]

${\displaystyle A=\lim _{n\rightarrow \infty }{\frac {H(n)}{n^{{\tfrac {n^{2}}{2}}+{\tfrac {n}{2}}+{\tfrac {1}{12}}}\,e^{-{\tfrac {n^{2}}{4}}}}}}$

where ${\displaystyle H(n)}$ is the hyperfactorial:$${\displaystyle H(n)=\prod _{i=1}^{n}i^{i}=1^{1}\cdot 2^{2}\cdot 3^{3}\cdot {...}\cdot n^{n}}$$An analogous limit, presenting a similarity between ${\displaystyle A}$ and ${\displaystyle {\sqrt {2\pi }}}$, is given by Stirling's formula as:

${\displaystyle {\sqrt {2\pi }}=\lim _{n\to \infty }{\frac {n!}{n^{n+{\frac {1}{2}}}\,e^{-n}}}}$

with$${\displaystyle n!=\prod _{i=1}^{n}i=1\cdot 2\cdot 3\cdot {...}\cdot n}$$which shows that just as π is obtained from approximation of the factorials, A is obtained from the approximation of the hyperfactorials.

Just as the factorials can be extended to the complex numbers by the gamma function such that ${\displaystyle \Gamma (n)=(n-1)!}$ for positive integers n, the hyperfactorials can be extended by the K-function^[3] with ${\displaystyle K(n)=H(n-1)}$ also for positive integers n, where:

${\displaystyle K(z)=(2\pi )^{-{\frac {z-1}{2}}}\exp \left[{\binom {z}{2}}+\int _{0}^{z-1}\ln \Gamma (t+1)\,dt\right]}$

This gives:^[1]

${\displaystyle A=\lim _{n\rightarrow \infty }{\frac {K(n+1)}{n^{{\tfrac {n^{2}}{2}}+{\tfrac {n}{2}}+{\tfrac {1}{12}}}\,e^{-{\tfrac {n^{2}}{4}}}}}}$.

A related function is the Barnes G-function which is given by

${\displaystyle G(n)={\frac {(\Gamma (n))^{n-1}}{K(n)}}}$

and for which a similar limit exists:^[2]

${\displaystyle {\frac {1}{A}}=\lim _{n\rightarrow \infty }{\frac {G(n+1)}{\left(2\pi \right)^{\frac {n}{2}}n^{{\frac {n^{2}}{2}}-{\frac {1}{12}}}e^{-{\frac {3n^{2}}{4}}+{\frac {1}{12}}}}}}$.

The Glaisher-Kinkelin constant also appears in the evaluation of the K-function and Barnes-G function at half and quarter integer values such as:^[1][4]

${\displaystyle K(1/2)={\frac {A^{3/2}}{2^{1/24}e^{1/8}}}}$

${\displaystyle K(1/4)=A^{9/8}\exp \left({\frac {G}{4\pi }}-{\frac {3}{32}}\right)}$

${\displaystyle G(1/2)={\frac {2^{1/24}e^{1/8}}{A^{3/2}\pi ^{1/4}}}}$

${\displaystyle G(1/4)={\frac {1}{2^{9/16}A^{9/8}\pi ^{3/16}\varpi ^{3/8}}}\exp \left({\frac {3}{32}}-{\frac {G}{4\pi }}\right)}$

with ${\displaystyle G}$ being Catalan's constant and ${\displaystyle \varpi ={\frac {\Gamma (1/4)^{2}}{2{\sqrt {2\pi }}}}}$ being the lemniscate constant.

Similar to the gamma function, there exists a multiplication formula for the K-Function. It involves Glaisher's constant:^[5]

${\displaystyle \prod _{j=1}^{n-1}K\left({\frac {j}{n}}\right)=A^{\frac {n^{2}-1}{n}}n^{-{\frac {1}{12n}}}e^{\frac {1-n^{2}}{12n}}}$

The logarithm of G(z + 1) has the following asymptotic expansion, as established by Barnes:^[6]

${\displaystyle \ln G(z+1)={\frac {z^{2}}{2}}\ln z-{\frac {3z^{2}}{4}}+{\frac {z}{2}}\ln 2\pi -{\frac {1}{12}}\ln z+\left({\frac {1}{12}}-\ln A\right)+\sum _{k=1}^{N}{\frac {B_{2k+2}}{4k\left(k+1\right)z^{2k}}}+O\left({\frac {1}{z^{2N+2}}}\right)}$

The Glaisher-Kinkelin constant is related to the derivatives of the Euler-constant function:^[5][7]

${\displaystyle \gamma '(-1)={\frac {11}{6}}\ln 2+6\ln A-{\frac {3}{2}}\ln \pi -1}$

${\displaystyle \gamma ''(-1)={\frac {10}{3}}\ln 2+24\ln A-4\ln \pi -{\frac {7\zeta (3)}{2\pi ^{2}}}-{\frac {13}{4}}}$

${\displaystyle A}$ also is related to the Lerch transcendent:^[8]

${\displaystyle {\frac {\partial \Phi }{\partial s}}(-1,-1,1)=3\ln A-{\frac {1}{3}}\ln 2-{\frac {1}{4}}}$

Glaisher's constant may be used to give values of the derivative of the Riemann zeta function as closed form expressions, such as:^[2][9]

${\displaystyle \zeta '(-1)={\frac {1}{12}}-\ln A}$

${\displaystyle \zeta '(2)={\frac {\pi ^{2}}{6}}\left(\gamma +\ln 2\pi -12\ln A\right)}$

where γ is the Euler–Mascheroni constant.

The above formula for ${\displaystyle \zeta '(2)}$ gives the following series:^[2]

${\displaystyle \sum _{k=2}^{\infty }{\frac {\ln k}{k^{2}}}={\frac {\pi ^{2}}{6}}\left(12\ln A-\gamma -\ln 2\pi \right)}$

which directly leads to the following product found by Glaisher:

${\displaystyle \prod _{k=1}^{\infty }k^{\frac {1}{k^{2}}}=\left({\frac {A^{12}}{2\pi e^{\gamma }}}\right)^{\frac {\pi ^{2}}{6}}}$

Similarly it is

${\displaystyle \sum _{k\geq 3}^{k{\text{ odd}}}{\frac {\ln k}{k^{2}}}={\frac {\pi ^{2}}{24}}\left(36\ln A-3\gamma -\ln 16\pi ^{3}\right)}$

which gives:

${\displaystyle \prod _{k\geq 3}^{k{\text{ odd}}}k^{\frac {1}{k^{2}}}=\left({\frac {A^{36}}{16\pi ^{3}e^{3\gamma }}}\right)^{\frac {\pi ^{2}}{24}}}$

An alternative product formula, defined over the prime numbers, reads:^[10]

${\displaystyle \prod _{p{\text{ prime}}}p^{\frac {1}{p^{2}-1}}={\frac {A^{12}}{2\pi e^{\gamma }}},}$

Another product is given by:^[5]

${\displaystyle \prod _{k=1}^{\infty }\left({\frac {en^{n}}{(n+1)^{n}}}\right)^{(-1)^{n-1}}={\frac {2^{1/6}e{\sqrt {\pi }}}{A^{6}}}}$

A series involving the cosine integral is:^[11]

${\displaystyle \sum _{k=1}^{\infty }{\frac {{\text{Ci}}(2k\pi )}{k^{2}}}={\frac {\pi ^{2}}{2}}(4\ln A-1)}$

Helmut Hasse gave another series representation for the logarithm of Glaisher's constant, following from a series for the Riemann zeta function:^[8]

${\displaystyle \ln A={\frac {1}{8}}-{\frac {1}{2}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+1)^{2}\ln(k+1)}$

The following are some definite integrals involving Glaisher's constant:^[1]

${\displaystyle \int _{0}^{\infty }{\frac {x\ln x}{e^{2\pi x}-1}}\,dx={\frac {1}{24}}-{\frac {1}{2}}\ln A}$

${\displaystyle \int _{0}^{\frac {1}{2}}\ln \Gamma (x)\,dx={\frac {3}{2}}\ln A+{\frac {5}{24}}\ln 2+{\frac {1}{4}}\ln \pi }$

the latter being a special case of:^[12]

${\displaystyle \int _{0}^{z}\ln \Gamma (x)\,dx={\frac {z(1-z)}{2}}+{\frac {z}{2}}\ln 2\pi +z\ln \Gamma (z)-\ln G(1+z)}$

We further have:^[13]$${\displaystyle \int _{0}^{\infty }{\frac {(1-e^{-x/2})(x\coth {\tfrac {x}{2}}-2)}{x^{3}}}dx=3\ln A-{\frac {1}{3}}\ln 2-{\frac {1}{8}}}$$and$${\displaystyle \int _{0}^{\infty }{\frac {(8-3x)e^{x}-8e^{x/2}-x}{4x^{2}e^{x}(e^{x}-1)}}dx=3\ln A-{\frac {7}{12}}\ln 2+{\frac {1}{2}}\ln \pi -1}$$A double integral is given by:^[8]

${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {-x}{(1+xy)^{2}\ln xy}}dxdy=6\ln A-{\frac {1}{6}}\ln 2-{\frac {1}{2}}\ln \pi -{\frac {1}{2}}}$

The Glaisher-Kinkelin constant can be viewed as the first constant in a sequence of infinitely many so-called generalized Glaisher constants or Bendersky constants.^[1] They emerge from studying the following product:$${\displaystyle \prod _{m=1}^{n}m^{m^{k}}=1^{1^{k}}\cdot 2^{2^{k}}\cdot 3^{3^{k}}\cdot {...}\cdot n^{n^{k}}}$$Setting ${\displaystyle k=0}$ gives the factorial ${\displaystyle n!}$, while choosing ${\displaystyle k=1}$ gives the hyperfactorial ${\displaystyle H(n)}$.

Defining the following function$${\displaystyle P_{k}(n)=\left({\frac {n^{k+1}}{k+1}}+{\frac {n^{k}}{2}}+{\frac {B_{k+1}}{k+1}}\right)\ln n-{\frac {n^{k+1}}{(k+1)^{2}}}+k!\sum _{j=1}^{k-1}{\frac {B_{j+1}}{(j+1)!}}{\frac {n^{k-j}}{(k-j)!}}\left(\ln n+\sum _{i=1}^{j}{\frac {1}{k-i+1}}\right)}$$with the Bernoulli numbers ${\displaystyle B_{k}}$ (and using ${\displaystyle B_{1}=0}$), one may approximate the above products asymptotically via ${\displaystyle \exp({P_{k}(n)})}$.

For ${\displaystyle k=0}$ we get Stirling's approximation without the factor ${\displaystyle {\sqrt {2\pi }}}$ as ${\displaystyle \exp({P_{0}(n)})=n^{n+{\frac {1}{2}}}e^{-n}}$.

For ${\displaystyle k=1}$ we obtain ${\displaystyle \exp({P_{1}(n)})=n^{{\tfrac {n^{2}}{2}}+{\tfrac {n}{2}}+{\tfrac {1}{12}}}\,e^{-{\tfrac {n^{2}}{4}}}}$, similar as in the limit definition of ${\displaystyle A}$.

This leads to the following definition of the generalized Glaisher constants:

${\displaystyle A_{k}:=\lim _{n\rightarrow \infty }\left(e^{-P_{k}(n)}\prod _{m=1}^{n}m^{m^{k}}\right)}$

which may also be written as:

${\displaystyle \ln A_{k}:=\lim _{n\rightarrow \infty }\left(-P_{k}(n)+\sum _{m=1}^{n}{m^{k}}\ln m\right)}$

This gives ${\displaystyle A_{0}={\sqrt {2\pi }}}$ and ${\displaystyle A_{1}=A}$ and in general:^[1][14][15]

${\displaystyle A_{k}=\exp \left({\frac {B_{k+1}}{k+1}}H_{k}-\zeta '(-k)\right)}$

with the harmonic numbers ${\displaystyle H_{k}}$ and ${\displaystyle H_{0}=0}$.

Because of the formula

${\displaystyle \zeta '(-2m)=(-1)^{m}{\frac {(2m)!}{2(2\pi )^{2m}}}\zeta (2m+1)}$

for ${\displaystyle m>0}$, there exist closed form expressions for ${\displaystyle A_{k}}$ with even ${\displaystyle k=2m}$ in terms of the values of the Riemann zeta function such as:^[1]

${\displaystyle A_{2}=\exp \left({\frac {\zeta (3)}{4\pi ^{2}}}\right)}$

${\displaystyle A_{4}=\exp \left(-{\frac {3\zeta (5)}{4\pi ^{4}}}\right)}$

For odd ${\displaystyle k=2m-1}$ one can express the constants ${\displaystyle A_{k}}$ in terms of the derivative of the Riemann zeta function such as:

${\displaystyle A_{1}=\exp \left(-{\frac {\zeta '(2)}{2\pi ^{2}}}+{\frac {\gamma +\ln 2\pi }{12}}\right)}$

${\displaystyle A_{3}=\exp \left({\frac {3\zeta '(4)}{4\pi ^{4}}}-{\frac {\gamma +\ln 2\pi }{120}}\right)}$

The numerical values of the first few generalized Glaisher constants are given below:

• Hyperfactorial
• Superfactorial
• Stirling's approximation
• List of mathematical constants

• The Glaisher–Kinkelin constant to 20,000 decimal places