# Glaisher

is Glaisher's constant with numerical value .

# Details • Mathematical constant treated as numeric by NumericQ and as a constant by D.
• Glaisher can be evaluated to any numerical precision using N.
• Glaisher's constant A satisfies , where is the Riemann zeta function.

# Background & Context

• Glaisher is the symbol representing Glaisher's constant , also known as the GlaisherKinkelin constant. Glaisher has a number of equivalent definitions throughout mathematics but is most commonly defined as the constant that satisfies , where is the Riemann zeta function Zeta, is its derivative evaluated at , and Log is the natural logarithm. Glaisher has a numerical value . Glaisher arises in mathematical computations including sums, products, and integrals but is especially prominent in sums and integrals involving Gamma and Zeta functions.
• When Glaisher is used as a symbol, it is propagated as an exact quantity. Expansion and simplification of complicated expressions involving Glaisher may require use of functions such as FunctionExpand and FullSimplify.
• It is not currently known if Glaisher is rational (meaning it can be expressed as a ratio of integers), algebraic (meaning it is the root of some integer polynomial), or normal (meaning the digits in its base- expansion are equally distributed) to any base.
• Glaisher can be evaluated to arbitrary numerical precision using N. However, no efficient formulas for computing large numbers of its digits are currently known. RealDigits can be used to return a list of digits of Glaisher and ContinuedFraction to obtain terms of its continued fraction expansion.

# Examples

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## Basic Examples(1)

Evaluate to any precision:

## Scope(2)

Do an exact numerical computation:

## Applications(5)

Derivatives of zeta functions:

Integrals containing gamma functions:

Get Glaisher from an infinite product:

Get Glaisher from infinite sums:

Obtains Glaisher from a limit:

## Properties & Relations(1)

Various symbolic relations are automatically used:

## Neat Examples(1)

Terms in the continued fraction: