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In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a
mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
, related to the -function and the Barnes -function. The constant appears in a number of
sums In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
and
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s, especially those involving
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
s and zeta functions. It is named after
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s
James Whitbread Lee Glaisher James Whitbread Lee Glaisher FRS FRSE FRAS (5 November 1848, Lewisham – 7 December 1928, Cambridge), son of James Glaisher and Cecilia Glaisher, was a prolific English mathematician and astronomer. His large collection of (mostly) Engli ...
and Hermann Kinkelin. Its approximate value is: : = ...   . The Glaisher–Kinkelin constant can be given by the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
: :A=\lim_ \frac where is the
hyperfactorial In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n is the product of the numbers of the form x^x from 1^1 to n^n. Definition The hyperfactorial of a positive integer n is the product of the numbers ...
. This formula displays a similarity between and which is perhaps best illustrated by noting
Stirling's formula In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less ...
: :\sqrt=\lim_ \frac which shows that just as is obtained from approximation of the
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) ...
s, 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 In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
is: :A=\lim_ \frac. The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as: :\zeta'(-1)=\tfrac-\ln A : \sum_^\infty \frac=-\zeta'(2)=\frac 6 \left( 12 \ln A - \gamma-\ln 2\pi \right) where is the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural ...
. The latter formula leads directly to the following product found by Glaisher: : \prod_^\infty k^\frac = \left(\frac \right)^\frac An alternative product formula, defined over the
prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, reads : \prod_^\infty p_k^\frac = \frac, where denotes the th
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
. The following are some integrals that involve this constant: : \int_0^\frac12 \ln\Gamma(x) \, dx = \tfrac 3 2 \ln A+\frac 5 \ln 2+\tfrac 1 4 \ln \pi : \int_0^\infty \frac \, dx = \tfrac 1 2 \zeta'(-1) = \tfrac 1 -\tfrac 1 2 \ln A A series representation for this constant follows from a series for the Riemann zeta function given by
Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory a ...
. : \ln A=\tfrac 1 8 - \tfrac 1 2 \sum_^\infty \frac 1 \sum_^n (-1)^k \binom n k (k+1)^2 \ln(k+1)


References

* * (Provides a variety of relationships.) * *


External links


The Glaisher–Kinkelin constant to 20,000 decimal places
{{DEFAULTSORT:Glaisher-Kinkelin constant Mathematical constants Number theory Glaisher family