A = 1.28242712910062263687... (sequence A074962 in the OEIS).
The Glaisher–Kinkelin constant A can be given by the limit:
where H(n) = Πn k=1kk is the hyperfactorial. This formula displays a similarity between A and π which is perhaps best illustrated by noting Stirling's formula:
which shows that just as π is obtained from approximation of the factorials, A can also be obtained from a similar approximation to the hyperfactorials.
An equivalent definition for A involving the Barnes G-function, given by G(n) = Πn−2 k=1k! = [Γ(n)]n−1/K(n) where Γ(n) is the gamma function is:
.
The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
The following are some integrals that involve this constant:
A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.
Referencesedit
^Van Gorder, Robert A. (2012). "Glaisher-Type Products over the Primes". International Journal of Number Theory. 08 (2): 543–550. doi:10.1142/S1793042112500297.
Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal. 16 (3): 247–270. arXiv:math.NT/0506319. doi:10.1007/s11139-007-9102-0. S2CID 14910435.
Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". Ramanujan Journal. 16 (3): 247–270. arXiv:math/0506319. doi:10.1007/s11139-007-9102-0. S2CID 14910435. (Provides a variety of relationships.)
glaisher, kinkelin, constant, mathematics, glaisher, constant, typically, denoted, mathematical, constant, related, function, barnes, function, constant, appears, number, sums, integrals, especially, those, involving, gamma, functions, zeta, functions, named, . In mathematics the Glaisher Kinkelin constant or Glaisher s constant typically denoted A is a mathematical constant related to the K function and the Barnes G 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 A 1 282427 129 100 622 636 87 sequence A074962 in the OEIS The Glaisher Kinkelin constant A can be given by the limit A limn H n nn22 n2 112e n24 displaystyle A lim n rightarrow infty frac H n n frac n 2 2 frac n 2 frac 1 12 e frac n 2 4 where H n Pnk 1 kk is the hyperfactorial This formula displays a similarity between A and p which is perhaps best illustrated by noting Stirling s formula 2p limn n nn 12e n displaystyle sqrt 2 pi lim n to infty frac n n n frac 1 2 e n which shows that just as p is obtained from approximation of the factorials A can also be obtained from a similar approximation to the hyperfactorials An equivalent definition for A involving the Barnes G function given by G n Pn 2k 1 k G n n 1 K n where G n is the gamma function is A limn 2p n2nn22 112e 3n24 112G n 1 displaystyle A lim n rightarrow infty frac left 2 pi right frac n 2 n frac n 2 2 frac 1 12 e frac 3n 2 4 frac 1 12 G n 1 The Glaisher Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function such as z 1 112 ln A displaystyle zeta 1 tfrac 1 12 ln A k 2 ln kk2 z 2 p26 12ln A g ln 2p displaystyle sum k 2 infty frac ln k k 2 zeta 2 frac pi 2 6 left 12 ln A gamma ln 2 pi right where g is the Euler Mascheroni constant The latter formula leads directly to the following product found by Glaisher k 1 k1k2 A122peg p26 displaystyle prod k 1 infty k frac 1 k 2 left frac A 12 2 pi e gamma right frac pi 2 6 An alternative product formula defined over the prime numbers reads 1 k 1 pk1pk2 1 A122peg displaystyle prod k 1 infty p k frac 1 p k 2 1 frac A 12 2 pi e gamma where pk denotes the k th prime number The following are some integrals that involve this constant 012ln G x dx 32ln A 524ln 2 14ln p displaystyle int 0 frac 1 2 ln Gamma x dx tfrac 3 2 ln A frac 5 24 ln 2 tfrac 1 4 ln pi 0 xln xe2px 1dx 12z 1 124 12ln A displaystyle int 0 infty frac x ln x e 2 pi x 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 18 12 n 0 1n 1 k 0n 1 k nk k 1 2ln k 1 displaystyle ln A tfrac 1 8 tfrac 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 References edit Van Gorder Robert A 2012 Glaisher Type Products over the Primes International Journal of Number Theory 08 2 543 550 doi 10 1142 S1793042112500297 Guillera Jesus Sondow Jonathan 2008 Double integrals and infinite products for some classical constants via analytic continuations of Lerch s transcendent The Ramanujan Journal 16 3 247 270 arXiv math NT 0506319 doi 10 1007 s11139 007 9102 0 S2CID 14910435 Guillera Jesus Sondow Jonathan 2008 Double integrals and infinite products for some classical constants via analytic continuations of Lerch s transcendent Ramanujan Journal 16 3 247 270 arXiv math 0506319 doi 10 1007 s11139 007 9102 0 S2CID 14910435 Provides a variety of relationships Weisstein Eric W Glaisher Kinkelin Constant MathWorld Weisstein Eric W Riemann Zeta Function MathWorld External links editThe Glaisher Kinkelin constant to 20 000 decimal places Retrieved from https en wikipedia org w index php title Glaisher Kinkelin constant amp oldid 1135603101, wikipedia, wiki, book, books, library,