Glaisher–Kinkelin constant explained

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

Its approximate value is:

= ...   .

The Glaisher–Kinkelin constant can be given by the limit:

A = \lim _{n → infty} (K(n +1 ))/(n ^{(n ^{2})/(+ (n)/(2) + (1)/(12) 2)} e ^{- (n ^{2})/(4)}) where is the hyperfactorial. This formula displays a similarity between and which is perhaps best illustrated by noting Stirling's formula:\sqrt{ 2 \pi} = \lim _{n} (n!)/(n ^{n + (12)/()} e ^{-n}) which shows that just as is obtained from approximation of the factorials, can also be obtained from a similar approximation to the hyperfactorials.

An equivalent definition for involving the Barnes -function, given by where is the gamma function is:

A = \lim _{n → infty} (\left( 2 \pi\right) ^{(n)/(2)} n ^{(n ^{2})/(- (1)/(12) 2)} e ^{- (3 n ^{2})/(+ (1)/(12) 4)})/(G(n +1 )) .The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:\zeta'( -1 ) = \tfrac{ 1 }{ 12 } -ln A \sum ^{inf ty} _{k =2} (ln k)/(=- \zeta'( 2 ) = k ^{2}) (\pi ^{2})/(6) \left( 12 ln A - \gamma -ln 2 \pi \right)

where is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher:

\prod ^{inf ty} _{k =1} k ^{(1)/(k ^{2})} = \left( (A ^{12})/(2 \pi e ^{\gamma}) \right) ^{(\pi ^{2})/(6)}

An alternative product formula, defined over the prime numbers, reads ^[1]

\prod ^{inf ty} _{k =1} p ^{(1)/(p ^{2-1} _{k})} _{k} = (A ^{12})/(2 \pi e ^{\gamma}) ,

where denotes the th prime number.

The following are some integrals that involve this constant:

\int ^{(12)/(ln \Gamma(x))} _{0} dx = \tfrac 3 2 ln A + (5)/(24) ln 2+ \tfrac 1 4 ln \pi \int ^{inf ty} _{0} (x ln x)/(e ^{2} -1) dx = \tfrac 1 2 \zeta'( -1 ) = \tfrac 1 { 24 } - \tfrac 1 2 ln A

A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.

ln A = \tfrac 1 8 - \tfrac 1 2 \sum ^{inf ty} _{n =0} (1)/(n +1) \sum ^{n} _{k =0} ( -1 ) ^{k} \binom n k (k +1 ) ^{2} ln (k +1 )

References

1. 10.1142/S1793042112500297. Glaisher-Type Products over the Primes. International Journal of Number Theory. 08. 2. 543–550. 2012. Van Gorder. Robert A..

• Jesus. Guillera. Jonathan. Sondow. math.NT/0506319. Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent. 2008. 10.1007/s11139-007-9102-0. 16. 3. The Ramanujan Journal. 247–270. 14910435.
• Jesus. Guillera. Jonathan. Sondow. Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent. Ramanujan Journal. 16. 3. 10.1007/s11139-007-9102-0. 2008. 247 - 270. math/0506319. 14910435. (Provides a variety of relationships.)

External links

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