Euler–Mascheroni constant


The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.
It is defined as the limiting difference between the harmonic series and the natural logarithm:
Here, represents the floor function.
The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:
Binary
Decimal
Hexadecimal
Continued fraction

Source:

History

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes. Euler used the notations and for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations and for the constant. The notation 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. For example, the German mathematician Carl Anton Bretschneider used the notation in 1835 and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842

Appearances

The Euler–Mascheroni constant appears, among other places, in the following :
The number has not been proved algebraic or transcendental. In fact, it is not even known whether is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if is rational, its denominator must be greater than 10244663. The ubiquity of revealed by the large number of equations below makes the irrationality of a major open question in mathematics. Also see.

Relation to gamma function

is related to the digamma function, and hence the derivative of the gamma function, when both functions are evaluated at 1. Thus:
This is equal to the limits:
Further limit results are :
A limit related to the beta function is

Relation to the zeta function

can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:
Other series related to the zeta function include:
The error term in the last equation is a rapidly decreasing function of. As a result, the formula is well-suited for efficient computation of the constant to high precision.
Other interesting limits equaling the Euler–Mascheroni constant are the antisymmetric limit :
and de la Vallée-Poussin's formula
where are ceiling brackets.
Closely related to this is the rational zeta series expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:
where is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers,. Expanding some of the terms in the Hurwitz zeta function gives:
where
can also be expressed as follows where is the Glaisher–Kinkelin constant:
can also be expressed as follows, which can be proven by expressing the zeta function as a Laurent series:

Integrals

equals the value of a number of definite integrals:
where is the fractional harmonic number.
Definite integrals in which appears include:
One can express using a special case of Hadjicostas's formula as a double integral and with equivalent series:
An interesting comparison by is the double integral and alternating series
It shows that may be thought of as an "alternating Euler constant".
The two constants are also related by the pair of series
where and are the number of 1s and 0s, respectively, in the base 2 expansion of.
We have also Catalan's 1875 integral

Series expansions

In general,
for any. However, the rate of convergence of this expansion depends significantly on. In particular, exhibits much more rapid convergence than the conventional expansion . This is because
while
Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.
Euler showed that the following infinite series approaches :
The series for is equivalent to a series Nielsen found in 1897 :
In 1910, Vacca found the closely related series
where is the logarithm to base 2 and is the floor function.
In 1926 he found a second series:
From the Malmsten–Kummer expansion for the logarithm of the gamma function we get:
An important expansion for Euler's constant is due to Fontana and Mascheroni
where are Gregory coefficients This series is the special case of the expansions
convergent for
A similar series with the Cauchy numbers of the second kind is
Blagouchine found an interesting generalisation of the Fontana-Mascheroni series
where are the Bernoulli polynomials of the second kind, which are defined by the generating function
For any rational this series contains rational terms only. For example, at, it becomes
see and. Other series with the same polynomials include these examples:
and
where is the gamma function.
A series related to the Akiyama-Tanigawa algorithm is
where are the Gregory coefficients of the second order.

Series of prime numbers:

Asymptotic expansions

equals the following asymptotic formulas :
The third formula is also called the Ramanujan expansion.
derived closed-form expressions for the sums of errors of these approximations. He showed that :

Exponential

The constant is important in number theory. Some authors denote this quantity simply as. equals the following limit, where is the th prime number:
This restates the third of Mertens' theorems. The numerical value of is:
Other infinite products relating to include:
These products result from the Barnes -function.
In addition,
where the th factor is the th root of
This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using hypergeometric functions.

Continued fraction

The continued fraction expansion of is of the form , which has no apparent pattern. The continued fraction is known to have at least 475,006 terms, and it has infinitely many terms if and only if is irrational.

Generalizations

Euler's generalized constants are given by
for, with as the special case . This can be further generalized to
for some arbitrary decreasing function. For example,
gives rise to the Stieltjes constants, and
gives
where again the limit
appears.
A two-dimensional limit generalization is the Masser–Gramain constant.
Euler–Lehmer constants are given by summation of inverses of numbers in a common
modulo class :
The basic properties are
and if then

Published digits

Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. 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.
DateDecimal digitsAuthorSources
17345Leonhard Euler
173515Leonhard Euler
178116Leonhard Euler
179032Lorenzo Mascheroni, with 20-22 and 31-32 wrong
180922Johann G. von Soldner
181122Carl Friedrich Gauss
181240Friedrich Bernhard Gottfried Nicolai
185734Christian Fredrik Lindman
186141Ludwig Oettinger
186749William Shanks
187199James W.L. Glaisher
1871101William Shanks
1877262J. C. Adams
1952328John William Wrench Jr.
1961Helmut Fischer and Karl Zeller
1962Donald Knuth
1962Dura W. Sweeney
1973William A. Beyer and Michael S. Waterman
1977Richard P. Brent
1980Richard P. Brent & Edwin M. McMillan
1993Jonathan Borwein
1999Patrick Demichel and Xavier Gourdon
March 13, 2009Alexander J. Yee & Raymond Chan,
December 22, 2013Alexander J. Yee,
March 15, 2016Peter Trueb
May 18, 2016Ron Watkins
August 23, 2017Ron Watkins
May 26, 2020Seungmin Kim & Ian Cutress