Euler's constant

The numerical value of Euler's constant, to 50 decimal places, is:[1] The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43), where he described it as "worthy of serious consideration".

[2][3] Euler initially calculated the constant's value to 6 decimal places.

The Italian mathematician Lorenzo Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 20th–22nd and 31st–32nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.

Other computations were done by Johann von Soldner in 1809, who used the notation H. The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time, perhaps because of the constant's connection to the gamma function.

[3] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835,[4] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.

[5] Euler's constant was also studied by the Indian mathematician Srinivasa Ramanujan who published one paper on it in 1917.

[6] David Hilbert mentioned the irrationality of γ as an unsolved problem that seems "unapproachable" and, allegedly, the English mathematician Godfrey Hardy offered to give up his Savilian Chair at Oxford to anyone who could prove this.

[2] Euler's constant appears frequently in mathematics, especially in number theory and analysis.

[7] Examples include, among others, the following places: (where '*' means that this entry contains an explicit equation): The number γ has not been proved algebraic or transcendental.

[3] In 2010, M. Ram Murty and N. Saradha showed that at most one of the Euler-Lehmer constants, i. e. the numbers of the form

Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10244663.

[3] Euler's constant is conjectured not to be an algebraic period,[3] but the values of its first 109 decimal digits seem to indicate that it could be a normal number.

[48] The simple continued fraction expansion of Euler's constant is given by:[49] which has no apparent pattern.

γ can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:

The error term in the last equation is a rapidly decreasing function of n. As a result, the formula is well-suited for efficient computation of the constant to high precision.

(an earlier, less-generalizable proof[63][64] by Ernesto Cesàro gives the first two terms of the series, with an error term): From Stirling's approximation[57][65] follows a similar series: The series of inverse triangular numbers also features in the study of the Basel problem[66][67] posed by Pietro Mengoli.

This identity appears in a formula used by Bernhard Riemann to compute roots of the zeta function,[68] where

plus the difference between Boya's expansion and the series of exact unit fractions

The integral on the second line of the equation stands for the Debye function value of +∞, which is m!ζ(m + 1).

One can express γ using a special case of Hadjicostas's formula as a double integral[39][70] with equivalent series:

An interesting comparison by Sondow[70] is the double integral and alternating series

Ramanujan, in his lost notebook gave a series that approaches γ[82]:

An important expansion for Euler's constant is due to Fontana and Mascheroni

convergent for k = 1, 2, ... A similar series with the Cauchy numbers of the second kind Cn is[80][84]

where ψn(a) are the Bernoulli polynomials of the second kind, which are defined by the generating function

Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations.

Its numerical value is:[87] eγ equals the following limit, where pn is the nth prime number:

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using hypergeometric functions.

, that occur in the Laurent series expansion of the Riemann zeta function: with

Euler–Lehmer constants are given by summation of inverses of numbers in a common modulo class:[43]