Euler's constant

Not to be confused with Euler's number, e ≈ 2.71828, the base of the natural logarithm.

Euler's constant
γ 0.57721...
General information
Type	Unknown
Fields	Number theory Analysis
History
Discovered	1734
By	Leonhard Euler
First mention	De Progressionibus harmonicis observationes
Named after	Leonhard Euler Lorenzo Mascheroni

The area of the blue region converges to Euler's constant.

Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:

$${\begin{aligned}\gamma &=\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)\\[5px]&=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,\mathrm {d} x.\end{aligned}}$$

Here, ⌊·⌋ represents the floor function.

The numerical value of Euler's constant, to 50 decimal places, is:

0.57721566490153286060651209008240243104215933593992... 

History

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". Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations C and O for the constant. 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. In 1790, he used the notations A and a for the constant. 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. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835, and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842. Euler's constant was also studied by the Indian mathematician Srinivasa Ramanujan who published one paper on it in 1917. 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.

Appearances

Euler's constant appears, among other places, in the following (where '*' means that this entry contains an explicit equation):

Analysis

• The Weierstrass product formula for the gamma function.
• The asymptotic expansion of the gamma function for small arguments.
• Calculations of the digamma function at rational values.
• The first term of the Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants.*
• The Laplace transform* of the natural logarithm.
• In the regularization/renormalization of the harmonic series as a finite value.
• Expressions involving the exponential integral.*
• A definition of the cosine integral.*
• Solution of the second kind to Bessel's equation.

Number theory

• An inequality for Euler's totient function.
• The growth rate of the divisor function.
• The calculation of the Meissel–Mertens constant.
• The third of Mertens' theorems.*
• Lower bounds to a prime gap.
• A formulation of the Riemann hypothesis.
• An approximation of the average number of divisors of all numbers from 1 to a given n.

In other fields

• In some formulations of Zipf's law
• The answer to the coupon collector's problem*
• The mean of the Gumbel distribution
• The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
• An upper bound on Shannon entropy in quantum information theory
• In dimensional regularization of Feynman diagrams in quantum field theory
• In the BCS equation on the critical temperature in Bardeen–Cooper–Schrieffer theory of superconductivity (BCS theory)*
• Fisher–Orr model for genetics of adaptation in evolutionary biology

Properties

Irrationality and transcendence

The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. The ubiquity of γ revealed by the large number of equations below and the fact that γ has been called the third most important mathematical constant after π and e makes the irrationality of γ a major open question in mathematics.

Unsolved problem in mathematics:

Is Euler's constant irrational? If so, is it transcendental?

(more unsolved problems in mathematics)

However, some progress has been made. In 1959 Andrei Shidlovsky proved that at least one of Euler's constant γ and the Gompertz constant δ is irrational; Tanguy Rivoal proved in 2012 that at least one of them is transcendental. Kurt Mahler showed in 1968 that the number ${\tfrac {\pi }{2}}{\tfrac {Y_{0}(2)}{J_{0}(2)}}-\gamma$ is transcendental (with $J_{\alpha }(x)$ and $Y_{\alpha }(x)$ being Bessel functions). It is known that the transcendence degree of the field $\mathbb {Q} (e,\gamma ,\delta )$ is at least two. In 2010 M. Ram Murty and N. Saradha showed that at most one of the Euler-Lehmer constants, a. i. the numbers of the form

$$\gamma (a,q)=\lim _{n\rightarrow \infty }\left(-{\frac {\log {(a+nq})}{q}}+\sum _{k=0}^{n}{\frac {1}{a+kq}}\right)$$

is algebraic, given that q ≥ 2 and 1 ≤ a < q; this family includes the special case γ(2,4) = ⁠γ/4⁠. In 2013 M. Ram Murty and A. Zaytseva found a different family containing γ, which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.

Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10²⁴⁴⁶⁶³. If e^γ is a rational number, then its denominator must be greater than 10¹⁵⁰⁰⁰.

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

Continued fraction

The simple continued fraction expansion of Euler's constant is given by:

$\gamma =0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{4+\dots }}}}}}}}}}}}}}$

which has no apparent pattern. It is known to have at least 16,695,000,000 terms, and it has infinitely many terms if and only if γ is irrational.

The Khinchin limits for $\pi$ (red), $\gamma$ (blue) and ${\sqrt[{3}]{2}}$ (green).

Numerical evidence suggests that both Euler's constant γ as well as the constant e^γ are among the numbers for which the geometric mean of their simple continued fraction terms converges to Khinchin's constant. Similarly, when $p_{n}/q_{n}$ are the convergents of their respective continued fractions, the limit $\lim _{n\to \infty }q_{n}^{1/n}$ appears to converge to Lévy's constant in both cases. However neither of these limits has been proven.

There also exists a generalized continued fraction for Euler's constant.

A good simple approximation of γ is given by the reciprocal of the square root of 3 or about 0.57735:

${\frac {1}{\sqrt {3}}}=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+\dots }}}}}}}}}}}}}}$

with the difference being about 1 in 7,429.

Formulas and identities

Relation to gamma function

γ is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:

$$-\gamma =\Gamma '(1)=\Psi (1).$$

This is equal to the limits:

$${\begin{aligned}-\gamma &=\lim _{z\to 0}\left(\Gamma (z)-{\frac {1}{z}}\right)\\&=\lim _{z\to 0}\left(\Psi (z)+{\frac {1}{z}}\right).\end{aligned}}$$

Further limit results are:

$${\begin{aligned}\lim _{z\to 0}{\frac {1}{z}}\left({\frac {1}{\Gamma (1+z)}}-{\frac {1}{\Gamma (1-z)}}\right)&=2\gamma \\\lim _{z\to 0}{\frac {1}{z}}\left({\frac {1}{\Psi (1-z)}}-{\frac {1}{\Psi (1+z)}}\right)&={\frac {\pi ^{2}}{3\gamma ^{2}}}.\end{aligned}}$$

A limit related to the beta function (expressed in terms of gamma functions) is

$${\begin{aligned}\gamma &=\lim _{n\to \infty }\left({\frac {\Gamma \left({\frac {1}{n}}\right)\Gamma (n+1)\,n^{1+{\frac {1}{n}}}}{\Gamma \left(2+n+{\frac {1}{n}}\right)}}-{\frac {n^{2}}{n+1}}\right)\\&=\lim \limits _{m\to \infty }\sum _{k=1}^{m}{m \choose k}{\frac {(-1)^{k}}{k}}\log {\big (}\Gamma (k+1){\big )}.\end{aligned}}$$

Relation to the zeta function

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

$${\begin{aligned}\gamma &=\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{m}}\\&=\log {\frac {4}{\pi }}+\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{2^{m-1}m}}.\end{aligned}}$$ The constant $\gamma$ can also be expressed in terms of the sum of the reciprocals of non-trivial zeros $\rho$ of the zeta function:

$\gamma =\log 4\pi +\sum _{\rho }{\frac {2}{\rho }}-2$

Other series related to the zeta function include:

$${\begin{aligned}\gamma &={\tfrac {3}{2}}-\log 2-\sum _{m=2}^{\infty }(-1)^{m}\,{\frac {m-1}{m}}{\big (}\zeta (m)-1{\big )}\\&=\lim _{n\to \infty }\left({\frac {2n-1}{2n}}-\log n+\sum _{k=2}^{n}\left({\frac {1}{k}}-{\frac {\zeta (1-k)}{n^{k}}}\right)\right)\\&=\lim _{n\to \infty }\left({\frac {2^{n}}{e^{2^{n}}}}\sum _{m=0}^{\infty }{\frac {2^{mn}}{(m+1)!}}\sum _{t=0}^{m}{\frac {1}{t+1}}-n\log 2+O\left({\frac {1}{2^{n}\,e^{2^{n}}}}\right)\right).\end{aligned}}$$

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.

Other interesting limits equaling Euler's constant are the antisymmetric limit:

$${\begin{aligned}\gamma &=\lim _{s\to 1^{+}}\sum _{n=1}^{\infty }\left({\frac {1}{n^{s}}}-{\frac {1}{s^{n}}}\right)\\&=\lim _{s\to 1}\left(\zeta (s)-{\frac {1}{s-1}}\right)\\&=\lim _{s\to 0}{\frac {\zeta (1+s)+\zeta (1-s)}{2}}\end{aligned}}$$

and the following formula, established in 1898 by de la Vallée-Poussin:

$$\gamma =\lim _{n\to \infty }{\frac {1}{n}}\,\sum _{k=1}^{n}\left(\left\lceil {\frac {n}{k}}\right\rceil -{\frac {n}{k}}\right)$$

where ⌈ ⌉ are ceiling brackets. This formula indicates that when taking any positive integer n and dividing it by each positive integer k less than n, the average fraction by which the quotient n/k falls short of the next integer tends to γ (rather than 0.5) as n tends to infinity.

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:

$$\gamma =\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n-\sum _{m=2}^{\infty }{\frac {\zeta (m,n+1)}{m}}\right),$$

where ζ(s, k) is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, H_n. Expanding some of the terms in the Hurwitz zeta function gives:

$$H_{n}=\log(n)+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-\varepsilon ,$$ where 0 < ε < ⁠1/252n⁶⁠.

γ can also be expressed as follows where A is the Glaisher–Kinkelin constant:

$$\gamma =12\,\log(A)-\log(2\pi )+{\frac {6}{\pi ^{2}}}\,\zeta '(2)$$

γ can also be expressed as follows, which can be proven by expressing the zeta function as a Laurent series:

$$\gamma =\lim _{n\to \infty }\left(-n+\zeta \left({\frac {n+1}{n}}\right)\right)$$

Relation to triangular numbers

Numerous formulations have been derived that express $\gamma$ in terms of sums and logarithms of triangular numbers. One of the earliest of these is a formula for the $n$th harmonic number attributed to Srinivasa Ramanujan where $\gamma$ is related to $\textstyle \ln 2T_{k}$ in a series that considers the powers of $\textstyle {\frac {1}{T_{k}}}$ (an earlier, less-generalizable proof by Ernesto Cesàro gives the first two terms of the series, with an error term):

${\begin{aligned}\gamma &=H_{u}-{\frac {1}{2}}\ln 2T_{u}-\sum _{k=1}^{v}{\frac {R(k)}{T_{u}^{k}}}-\Theta _{v}\,{\frac {R(v+1)}{T_{u}^{v+1}}}\end{aligned}}$

From Stirling's approximation follows a similar series:

$\gamma =\ln 2\pi -\sum _{k=2}^{n}{\frac {\zeta (k)}{T_{k}}}$

The series of inverse triangular numbers also features in the study of the Basel problem posed by Pietro Mengoli. Mengoli proved that $\textstyle \sum _{k=1}^{\infty }{\frac {1}{2T_{k}}}=1$, a result Jacob Bernoulli later used to estimate the value of $\zeta (2)$, placing it between $1$ and $\textstyle \sum _{k=1}^{\infty }{\frac {2}{2T_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{T_{k}}}=2$. This identity appears in a formula used by Bernhard Riemann to compute roots of the zeta function, where $\gamma$ is expressed in terms of the sum of roots $\rho$ plus the difference between Boya's expansion and the series of exact unit fractions $\textstyle \sum _{k=1}^{n}{\frac {1}{T_{k}}}$:

$\gamma -\ln 2=\ln 2\pi +\sum _{\rho }{\frac {2}{\rho }}-\sum _{k=1}^{n}{\frac {1}{T_{k}}}$

Integrals

γ equals the value of a number of definite integrals:

$${\begin{aligned}\gamma &=-\int _{0}^{\infty }e^{-x}\log x\,dx\\&=-\int _{0}^{1}\log \left(\log {\frac {1}{x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{x\cdot e^{x}}}\right)dx\\&=\int _{0}^{1}{\frac {1-e^{-x}}{x}}\,dx-\int _{1}^{\infty }{\frac {e^{-x}}{x}}\,dx\\&=\int _{0}^{1}\left({\frac {1}{\log x}}+{\frac {1}{1-x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{1+x^{k}}}-e^{-x}\right){\frac {dx}{x}},\quad k>0\\&=2\int _{0}^{\infty }{\frac {e^{-x^{2}}-e^{-x}}{x}}\,dx,\\&=\log {\frac {\pi }{4}}-\int _{0}^{\infty }{\frac {\log x}{\cosh ^{2}x}}\,dx,\\&=\int _{0}^{1}H_{x}\,dx,\\&={\frac {1}{2}}+\int _{0}^{\infty }\log \left(1+{\frac {\log \left(1+{\frac {1}{t}}\right)^{2}}{4\pi ^{2}}}\right)dt\\&=1-\int _{0}^{1}\{1/x\}dx\end{aligned}}$$ where H_x is the fractional harmonic number, and $\{1/x\}$ is the fractional part of $1/x$.

The third formula in the integral list can be proved in the following way:

$${\begin{aligned}&\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{xe^{x}}}\right)dx=\int _{0}^{\infty }{\frac {e^{-x}+x-1}{x[e^{x}-1]}}dx=\int _{0}^{\infty }{\frac {1}{x[e^{x}-1]}}\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m+1}}{(m+1)!}}dx\\[2pt]&=\int _{0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![e^{x}-1]}}dx=\sum _{m=1}^{\infty }\int _{0}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![e^{x}-1]}}dx=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}\int _{0}^{\infty }{\frac {x^{m}}{e^{x}-1}}dx\\[2pt]&=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}m!\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\sum _{n=1}^{\infty }{\frac {1}{n^{m+1}}}=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}\\[2pt]&=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}=\sum _{n=1}^{\infty }\left[{\frac {1}{n}}-\log \left(1+{\frac {1}{n}}\right)\right]=\gamma \end{aligned}}$$

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

Definite integrals in which γ appears include:

$${\begin{aligned}\int _{0}^{\infty }e^{-x^{2}}\log x\,dx&=-{\frac {(\gamma +2\log 2){\sqrt {\pi }}}{4}}\\\int _{0}^{\infty }e^{-x}\log ^{2}x\,dx&=\gamma ^{2}+{\frac {\pi ^{2}}{6}}\\\int _{0}^{\infty }{\frac {e^{-x}\log x}{e^{x}+1}}\,dx&={\frac {1}{2}}\log ^{2}2-\gamma \end{aligned}}$$

We also have Catalan's 1875 integral

$$\gamma =\int _{0}^{1}\left({\frac {1}{1+x}}\sum _{n=1}^{\infty }x^{2^{n}-1}\right)\,dx.$$

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

$${\begin{aligned}\gamma &=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-xy)\log xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\log {\frac {n+1}{n}}\right).\end{aligned}}$$

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

$${\begin{aligned}\log {\frac {4}{\pi }}&=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1+xy)\log xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left((-1)^{n-1}\left({\frac {1}{n}}-\log {\frac {n+1}{n}}\right)\right).\end{aligned}}$$

It shows that log ⁠4/π⁠ may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series

$${\begin{aligned}\gamma &=\sum _{n=1}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}\\\log {\frac {4}{\pi }}&=\sum _{n=1}^{\infty }{\frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}},\end{aligned}}$$

where N₁(n) and N₀(n) are the number of 1s and 0s, respectively, in the base 2 expansion of n.

Series expansions

In general,

$$\gamma =\lim _{n\to \infty }\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\ldots +{\frac {1}{n}}-\log(n+\alpha )\right)\equiv \lim _{n\to \infty }\gamma _{n}(\alpha )$$

for any α > −n. However, the rate of convergence of this expansion depends significantly on α. In particular, γ_n(1/2) exhibits much more rapid convergence than the conventional expansion γ_n(0). This is because

$${\frac {1}{2(n+1)}}<\gamma _{n}(0)-\gamma <{\frac {1}{2n}},$$

while

$${\frac {1}{24(n+1)^{2}}}<\gamma _{n}(1/2)-\gamma <{\frac {1}{24n^{2}}}.$$

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 γ: $$\gamma =\sum _{k=1}^{\infty }\left({\frac {1}{k}}-\log \left(1+{\frac {1}{k}}\right)\right).$$

The series for γ is equivalent to a series Nielsen found in 1897:

$$\gamma =1-\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k+1}}.$$

In 1910, Vacca found the closely related series

$${\begin{aligned}\gamma &=\sum _{k=1}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}\\[5pt]&={\tfrac {1}{2}}-{\tfrac {1}{3}}+2\left({\tfrac {1}{4}}-{\tfrac {1}{5}}+{\tfrac {1}{6}}-{\tfrac {1}{7}}\right)+3\left({\tfrac {1}{8}}-{\tfrac {1}{9}}+{\tfrac {1}{10}}-{\tfrac {1}{11}}+\cdots -{\tfrac {1}{15}}\right)+\cdots ,\end{aligned}}$$

where log₂ is the logarithm to base 2 and ⌊ ⌋ is the floor function.

This can be generalized to:

$$\gamma =\sum _{k=1}^{\infty }{\frac {\left\lfloor \log _{B}k\right\rfloor }{k}}\varepsilon (k)$$where:$$\varepsilon (k)={\begin{cases}B-1,&{\text{if }}B\mid n\\-1,&{\text{if }}B\nmid n\end{cases}}$$

In 1926 Vacca found a second series:

$${\begin{aligned}\gamma +\zeta (2)&=\sum _{k=2}^{\infty }\left({\frac {1}{\left\lfloor {\sqrt {k}}\right\rfloor ^{2}}}-{\frac {1}{k}}\right)\\[5pt]&=\sum _{k=2}^{\infty }{\frac {k-\left\lfloor {\sqrt {k}}\right\rfloor ^{2}}{k\left\lfloor {\sqrt {k}}\right\rfloor ^{2}}}\\[5pt]&={\frac {1}{2}}+{\frac {2}{3}}+{\frac {1}{2^{2}}}\sum _{k=1}^{2\cdot 2}{\frac {k}{k+2^{2}}}+{\frac {1}{3^{2}}}\sum _{k=1}^{3\cdot 2}{\frac {k}{k+3^{2}}}+\cdots \end{aligned}}$$

From the Malmsten–Kummer expansion for the logarithm of the gamma function we get:

$$\gamma =\log \pi -4\log \left(\Gamma ({\tfrac {3}{4}})\right)+{\frac {4}{\pi }}\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {\log(2k+1)}{2k+1}}.$$

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

$$\gamma =\log 2-\sum _{n=1}^{\infty }\sum _{k={\frac {3^{n-1}+1}{2}}}^{\frac {3^{n}-1}{2}}{\frac {2n}{(3k)^{3}-3k}}$$

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

$$\gamma =\sum _{n=1}^{\infty }{\frac {|G_{n}|}{n}}={\frac {1}{2}}+{\frac {1}{24}}+{\frac {1}{72}}+{\frac {19}{2880}}+{\frac {3}{800}}+\cdots ,$$ where G_n are Gregory coefficients. This series is the special case k = 1 of the expansions

$${\begin{aligned}\gamma &=H_{k-1}-\log k+\sum _{n=1}^{\infty }{\frac {(n-1)!|G_{n}|}{k(k+1)\cdots (k+n-1)}}&&\\&=H_{k-1}-\log k+{\frac {1}{2k}}+{\frac {1}{12k(k+1)}}+{\frac {1}{12k(k+1)(k+2)}}+{\frac {19}{120k(k+1)(k+2)(k+3)}}+\cdots &&\end{aligned}}$$

convergent for k = 1, 2, ...

A similar series with the Cauchy numbers of the second kind C_n is

$$\gamma =1-\sum _{n=1}^{\infty }{\frac {C_{n}}{n\,(n+1)!}}=1-{\frac {1}{4}}-{\frac {5}{72}}-{\frac {1}{32}}-{\frac {251}{14400}}-{\frac {19}{1728}}-\ldots$$

Blagouchine (2018) found an interesting generalisation of the Fontana–Mascheroni series

$$\gamma =\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2n}}{\Big \{}\psi _{n}(a)+\psi _{n}{\Big (}-{\frac {a}{1+a}}{\Big )}{\Big \}},\quad a>-1$$

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

$${\frac {z(1+z)^{s}}{\log(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(s),\qquad |z|<1.$$

For any rational a this series contains rational terms only. For example, at a = 1, it becomes

$$\gamma ={\frac {3}{4}}-{\frac {11}{96}}-{\frac {1}{72}}-{\frac {311}{46080}}-{\frac {5}{1152}}-{\frac {7291}{2322432}}-{\frac {243}{100352}}-\ldots$$ Other series with the same polynomials include these examples:

$$\gamma =-\log(a+1)-\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)}{n}},\qquad \Re (a)>-1$$

and

$$\gamma =-{\frac {2}{1+2a}}\left\{\log \Gamma (a+1)-{\frac {1}{2}}\log(2\pi )+{\frac {1}{2}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n+1}(a)}{n}}\right\},\qquad \Re (a)>-1$$

where Γ(a) is the gamma function.

A series related to the Akiyama–Tanigawa algorithm is

$$\gamma =\log(2\pi )-2-2\sum _{n=1}^{\infty }{\frac {(-1)^{n}G_{n}(2)}{n}}=\log(2\pi )-2+{\frac {2}{3}}+{\frac {1}{24}}+{\frac {7}{540}}+{\frac {17}{2880}}+{\frac {41}{12600}}+\ldots$$

where G_n(2) are the Gregory coefficients of the second order.

As a series of prime numbers:

$$\gamma =\lim _{n\to \infty }\left(\log n-\sum _{p\leq n}{\frac {\log p}{p-1}}\right).$$

Asymptotic expansions

γ equals the following asymptotic formulas (where H_n is the nth harmonic number):

• ${\textstyle \gamma \sim H_{n}-\log n-{\frac {1}{2n}}+{\frac {1}{12n^{2}}}-{\frac {1}{120n^{4}}}+\cdots }$ (Euler)
• ${\textstyle \gamma \sim H_{n}-\log \left({n+{\frac {1}{2}}+{\frac {1}{24n}}-{\frac {1}{48n^{2}}}+\cdots }\right)}$ (Negoi)
• ${\textstyle \gamma \sim H_{n}-{\frac {\log n+\log(n+1)}{2}}-{\frac {1}{6n(n+1)}}+{\frac {1}{30n^{2}(n+1)^{2}}}-\cdots }$ (Cesàro)

The third formula is also called the Ramanujan expansion.

Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations. He showed that (Theorem A.1):

$${\begin{aligned}\sum _{n=1}^{\infty }\log n+\gamma -H_{n}+{\frac {1}{2n}}&={\frac {\log(2\pi )-1-\gamma }{2}}\\\sum _{n=1}^{\infty }\log {\sqrt {n(n+1)}}+\gamma -H_{n}&={\frac {\log(2\pi )-1}{2}}-\gamma \\\sum _{n=1}^{\infty }(-1)^{n}{\Big (}\log n+\gamma -H_{n}{\Big )}&={\frac {\log \pi -\gamma }{2}}\end{aligned}}$$

Exponential

The constant e^γ is important in number theory. Its numerical value is:

1.78107241799019798523650410310717954916964521430343....

e^γ equals the following limit, where p_n is the nth prime number:

$$e^{\gamma }=\lim _{n\to \infty }{\frac {1}{\log p_{n}}}\prod _{i=1}^{n}{\frac {p_{i}}{p_{i}-1}}.$$

This restates the third of Mertens' theorems.

We further have the following product involving the three constants e, π and γ:

$${\frac {\pi ^{2}}{6e^{\gamma }}}=\lim _{n\to \infty }{\frac {1}{\log p_{n}}}\prod _{i=1}^{n}{\frac {p_{i}}{p_{i}+1}}.$$

Other infinite products relating to e^γ include:

$${\begin{aligned}{\frac {e^{1+{\frac {\gamma }{2}}}}{\sqrt {2\pi }}}&=\prod _{n=1}^{\infty }e^{-1+{\frac {1}{2n}}}\left(1+{\frac {1}{n}}\right)^{n}\\{\frac {e^{3+2\gamma }}{2\pi }}&=\prod _{n=1}^{\infty }e^{-2+{\frac {2}{n}}}\left(1+{\frac {2}{n}}\right)^{n}.\end{aligned}}$$

These products result from the Barnes G-function.

In addition,

$$e^{\gamma }={\sqrt {\frac {2}{1}}}\cdot {\sqrt[{3}]{\frac {2^{2}}{1\cdot 3}}}\cdot {\sqrt[{4}]{\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}}\cdot {\sqrt[{5}]{\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}}\cdots$$

where the nth factor is the (n + 1)th root of

$$\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}.$$

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

It also holds that

$${\frac {e^{\frac {\pi }{2}}+e^{-{\frac {\pi }{2}}}}{\pi e^{\gamma }}}=\prod _{n=1}^{\infty }\left(e^{-{\frac {1}{n}}}\left(1+{\frac {1}{n}}+{\frac {1}{2n^{2}}}\right)\right).$$

Published digits

Published decimal expansions of γ

Date	Decimal digits	Author	Sources
1734	5	Leonhard Euler
1735	15	Leonhard Euler
1781	16	Leonhard Euler
1790	32	Lorenzo Mascheroni, with 20–22 and 31–32 wrong
1809	22	Johann G. von Soldner
1811	22	Carl Friedrich Gauss
1812	40	Friedrich Bernhard Gottfried Nicolai
1861	41	Ludwig Oettinger
1867	49	William Shanks
1871	100	James W.L. Glaisher
1877	263	J. C. Adams
1952	328	John William Wrench Jr.
1961	1050	Helmut Fischer and Karl Zeller
1962	1271	Donald Knuth
1963	3566	Dura W. Sweeney
1973	4879	William A. Beyer and Michael S. Waterman
1977	20700	Richard P. Brent
1980	30100	Richard P. Brent & Edwin M. McMillan
1993	172000	Jonathan Borwein
1997	1000000	Thomas Papanikolaou
1998	7286255	Xavier Gourdon
1999	108000000	Patrick Demichel and Xavier Gourdon
March 13, 2009	29844489545	Alexander J. Yee & Raymond Chan
December 22, 2013	119377958182	Alexander J. Yee
March 15, 2016	160000000000	Peter Trueb
May 18, 2016	250000000000	Ron Watkins
August 23, 2017	477511832674	Ron Watkins
May 26, 2020	600000000100	Seungmin Kim & Ian Cutress
May 13, 2023	700000000000	Jordan Ranous & Kevin O'Brien
September 7, 2023	1337000000000	Andrew Sun

Generalizations

Stieltjes constants

Main article: Stieltjes constants

Euler's generalized constants abm(-$\alpha$) for α > 0.

Euler's generalized constants are given by

$$\gamma _{\alpha }=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k^{\alpha }}}-\int _{1}^{n}{\frac {1}{x^{\alpha }}}\,dx\right)$$

for 0 < α < 1, with γ as the special case α = 1. Extending for α > 1 gives:

$$\gamma _{\alpha }=\zeta (\alpha )-{\frac {1}{\alpha -1}}$$

with again the limit:

$$\gamma =\lim _{a\to 1}\left(\zeta (a)-{\frac {1}{a-1}}\right)$$

This can be further generalized to

$$c_{f}=\lim _{n\to \infty }\left(\sum _{k=1}^{n}f(k)-\int _{1}^{n}f(x)\,dx\right)$$

for some arbitrary decreasing function f. Setting

$$f_{n}(x)={\frac {(\log x)^{n}}{x}}$$

gives rise to the Stieltjes constants $\gamma _{n}$, that occur in the Laurent series expansion of the Riemann zeta function:

$\zeta (1+s)={\frac {1}{s}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\gamma _{n}s^{n}.$

with $\gamma _{0}=\gamma =0.577\dots$

n	approximate value of γn	OEIS
0	+0.5772156649015	A001620
1	−0.0728158454836	A082633
2	−0.0096903631928	A086279
3	+0.0020538344203	A086280
4	+0.0023253700654	A086281
100	−4.2534015717080 × 1017
1000	−1.5709538442047 × 10486

Euler-Lehmer constants

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

$$\gamma (a,q)=\lim _{x\to \infty }\left(\sum _{0<n\leq x \atop n\equiv a{\pmod {q}}}{\frac {1}{n}}-{\frac {\log x}{q}}\right).$$

The basic properties are

$${\begin{aligned}&\gamma (0,q)={\frac {\gamma -\log q}{q}},\\&\sum _{a=0}^{q-1}\gamma (a,q)=\gamma ,\\&q\gamma (a,q)=\gamma -\sum _{j=1}^{q-1}e^{-{\frac {2\pi aij}{q}}}\log \left(1-e^{\frac {2\pi ij}{q}}\right),\end{aligned}}$$

and if the greatest common divisor gcd(a,q) = d then

$$q\gamma (a,q)={\frac {q}{d}}\gamma \left({\frac {a}{d}},{\frac {q}{d}}\right)-\log d.$$

Masser-Gramain constant

A two-dimensional generalization of Euler's constant is the Masser-Gramain constant. It is defined as the following limiting difference:

$\delta =\lim _{n\to \infty }\left(-\log n+\sum _{k=2}^{n}{\frac {1}{\pi r_{k}^{2}}}\right)$

where $r_{k}$ is the smallest radius of a disk in the complex plane containing at least $k$ Gaussian integers.

The following bounds have been established: $1.819776<\delta <1.819833$.

See also

• Harmonic series
• Riemann zeta function
• Stieltjes constants
• Meissel-Mertens constant

References

• Bretschneider, Carl Anton (1837) [1835]. "Theoriae logarithmi integralis lineamenta nova". Crelle's Journal (in Latin). 17: 257–285.
• Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 978-0-691-09983-5.
• Lagarias, Jeffrey C. (2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 556. arXiv:1303.1856. doi:10.1090/s0273-0979-2013-01423-x. S2CID 119612431.

Further reading

• Borwein, Jonathan M.; David M. Bradley; Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function". Journal of Computational and Applied Mathematics. 121 (1–2): 11. Bibcode:2000JCoAM.121..247B. doi:10.1016/s0377-0427(00)00336-8. Derives γ as sums over Riemann zeta functions.
• Finch, Steven R. (2003). Mathematical Constants. Encyclopedia of Mathematics and its Applications. Vol. 94. Cambridge: Cambridge University Press. ISBN 0-521-81805-2.
• Gerst, I. (1969). "Some series for Euler's constant". Amer. Math. Monthly. 76 (3): 237–275. doi:10.2307/2316370. JSTOR 2316370.
• Glaisher, James Whitbread Lee (1872). "On the history of Euler's constant". Messenger of Mathematics. 1: 25–30. JFM 03.0130.01.
• Gourdon, Xavier; Sebah, P. (2002). "Collection of formulae for Euler's constant, γ".
• Gourdon, Xavier; Sebah, P. (2004). "The Euler constant: γ".
• Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (44): 339–360.
• Karatsuba, E.A. (2000). "On the computation of the Euler constant γ". Journal of Numerical Algorithms. 24 (1–2): 83–97. doi:10.1023/A:1019137125281. S2CID 21545868.
• Knuth, Donald (1997). The Art of Computer Programming, Vol. 1 (3rd ed.). Addison-Wesley. pp. 75, 107, 114, 619–620. ISBN 0-201-89683-4.
• Lehmer, D. H. (1975). "Euler constants for arithmetical progressions" (PDF). Acta Arith. 27 (1): 125–142. doi:10.4064/aa-27-1-125-142.
• Lerch, M. (1897). "Expressions nouvelles de la constante d'Euler". Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften. 42: 5.
• Mascheroni, Lorenzo (1790). Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur. Galeati, Ticini.
• Sondow, Jonathan (2002). "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant". Mathematica Slovaca. 59: 307–314. arXiv:math.NT/0211075. Bibcode:2002math.....11075S. doi:10.2478/s12175-009-0127-2. S2CID 16340929. with an Appendix by Sergey Zlobin

External links

• "Euler constant". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
• Weisstein, Eric W. "Euler–Mascheroni constant". MathWorld.
• Jonathan Sondow.
• Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
• Further formulae which make use of the constant: Gourdon and Sebah (2004).