Euler's constant

File:Gamma-area.svg

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:$$\begin{array}{rl}\gamma & =\underset{n\to \mathrm{\infty }}{\lim }(\sum _{k=1}^n\frac{1}{k}-\log n)\\ & =\underset{1}{\overset{\mathrm{\infty }}{\int }}(\frac{1}{\lfloor x\rfloor }-\frac{1}{x})\mathrm{d}x.\end{array}$$Here, ⌊·⌋ represents the floor function.

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

History

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Observations on harmonic progressions; Eneström Index 43), where he described it as "worthy of serious consideration".Template:Sfn 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.Template:Sfn For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835,Template:Sfn and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.Template:R 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 frequently in mathematics, especially in number theory and analysis. Examples include, among others, the following places: (where '*' means that this entry contains an explicit equation):

Analysis

• The Weierstrass product formula for the gamma function and the Barnes G-function.
• The asymptotic expansion of the gamma function, $\mathrm{\Gamma }(1/x)\sim x-\gamma$.
• Evaluations of the digamma function at rational values.
• The Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants.
• Values of the derivative of the Riemann zeta function and Dirichlet beta function.Template:Rp
• In connection to the Laplace and Mellin transform.
• In the regularization/renormalization of the harmonic series as a finite value.
• Expressions involving the exponential and logarithmic integral.*
• A definition of the cosine integral.*
• In relation to Bessel functions.
• Asymptotic expansions of modified Struve functions.
• In relation to other special functions.

Number theory

• An inequality for Euler's totient function.
• The growth rate of the divisor function.
• A formulation of the Riemann hypothesis.
• The third of Mertens' theorems.*
• The calculation of the Meissel–Mertens constant.
• Lower bounds to specific prime gaps.
• An approximation of the average number of divisors of all numbers from 1 to a given n.
• The Lenstra–Pomerance–Wagstaff conjecture on the frequency of Mersenne primes.
• An estimation of the efficiency of the euclidean algorithm.
• Sums involving the Möbius and von Mangolt function.
• Estimate of the divisor summatory function of the Dirichlet hyperbola method.

In other fields

• In some formulations of Zipf's law.
• The answer to the coupon collector's problem.*
• The mean of the Gumbel distribution.
• An approximation of the Landau 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.Template:R
• In dimensional regularization of Feynman diagrams in quantum field theory.
• In the BCS equation on the critical temperature in BCS theory of superconductivity.*
• Fisher–Orr model for genetics of adaptation in evolutionary biology.Template:R

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 Template:Var has been called the third most important mathematical constant after [[Pi|Template:Var]] and [[E (mathematical constant)|Template:Var]] makes the irrationality of γ a major open question in mathematics.Template:R

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;Template:R Tanguy Rivoal proved in 2012 that at least one of them is transcendental.Template:R Kurt Mahler showed in 1968 that the number $\frac{\pi }{2}\frac{Y_0(2)}{J_0(2)}-\gamma$ is transcendental, where $J_0$ and $Y_0$ are the usual Bessel functions.Template:RTemplate:Sfn It is known that the transcendence degree of the field $\mathbb{\mathbb{Q}}(e,\gamma ,\delta )$ is at least two.Template:Sfn

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 $$\gamma (a,q)=\underset{n\to \mathrm{\infty }}{\lim }(\sum _{k=0}^n\frac{1}{a+kq}-\frac{\log (a+nq)}{q})$$ is algebraic, if q ≥ 2 and 1 ≤ a < q; this family includes the special case γ(2,4) = γ/4.Template:SfnTemplate:R

Using the same approach, in 2013, M. Ram Murty and A. Zaytseva showed that the generalized Euler constants have the same property, Template:SfnTemplate:R where the generalized Euler constant are defined as $$\gamma (\mathrm{\Omega })=\underset{x\to \mathrm{\infty }}{\lim }(\sum _{n=1}^x\frac{1_{\mathrm{\Omega }}(n)}{n}-\log x\cdot \underset{x\to \mathrm{\infty }}{\lim }\frac{\sum _{n=1}^x{1}_{\mathrm{\Omega }}(n)}{x}),$$ where Template:Tmath is a fixed list of prime numbers, $1_{\mathrm{\Omega }}(n)=0$ if at least one of the primes in Template:Tmath is a prime factor of Template:Tmath, and $1_{\mathrm{\Omega }}(n)=1$ otherwise. In particular, Template:Tmath.

Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10²⁴⁴⁶⁶³.Template:R If Template:VarTemplate:I sup is a rational number, then its denominator must be greater than 10¹⁵⁰⁰⁰.Template:Sfn

Euler's constant is conjectured not to be an algebraic period,Template:Sfn 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:Template:R

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

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

File:KhinchinBeispiele.svg

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

Numerical evidence suggests that both Euler's constant Template:Var as well as the constant Template:VarTemplate:I sup 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 $\underset{n\to \mathrm{\infty }}{\lim }{q}_n^{1/n}$ 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 Template:Var is given by the reciprocal of the square root of 3 or about 0.57735:

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

with the difference being about 1 in 7,429.

Formulas and identities

Relation to gamma function

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

$$-\gamma ={\mathrm{\Gamma }}^{\prime }(1)=\psi (1).$$

This is equal to the limits:

$$\begin{array}{rl}-\gamma & =\underset{z\to 0}{\lim }(\mathrm{\Gamma }(z)-\frac{1}{z})\\ & =\underset{z\to 0}{\lim }(\psi (z)+\frac{1}{z}).\end{array}$$

Further limit results are:Template:R

$$\begin{array}{rl}\underset{z\to 0}{\lim }\frac{1}{z}(\frac{1}{\mathrm{\Gamma }(1+z)}-\frac{1}{\mathrm{\Gamma }(1-z)})& =2\gamma \\ \underset{z\to 0}{\lim }\frac{1}{z}(\frac{1}{\psi (1-z)}-\frac{1}{\psi (1+z)})& =\frac{{\pi }^2}{3{\gamma }^2}.\end{array}$$

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

$$\begin{array}{rl}\gamma & =\underset{n\to \mathrm{\infty }}{\lim }(\frac{\mathrm{\Gamma }\left(\frac{1}{n}\right)\mathrm{\Gamma }(n+1)n^{1+\frac{1}{n}}}{\mathrm{\Gamma }(2+n+\frac{1}{n})}-\frac{n^2}{n+1})\\ & =\lim {}_{m\to \mathrm{\infty }}\sum _{k=1}^m(\genfrac{}{}{0ex}{}{m}{k})\frac{(-1{)}^k}{k}\log (\mathrm{\Gamma }(k+1)).\end{array}$$

Relation to the zeta function

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

$$\begin{array}{rl}\gamma & =\sum _{m=2}^{\mathrm{\infty }}(-1{)}^m\frac{\zeta (m)}{m}\\ & =\log \frac{4}{\pi }+\sum _{m=2}^{\mathrm{\infty }}(-1{)}^m\frac{\zeta (m)}{2^{m-1}m}.\end{array}$$ 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:

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

The error term in the last equation is a rapidly decreasing function of Template:Mvar. 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:Template:R

$$\begin{array}{rl}\gamma & =\underset{s\to 1^{+}}{\lim }\sum _{n=1}^{\mathrm{\infty }}(\frac{1}{n^s}-\frac{1}{s^n})\\ & =\underset{s\to 1}{\lim }(\zeta (s)-\frac{1}{s-1})\\ & =\underset{s\to 0}{\lim }\frac{\zeta (1+s)+\zeta (1-s)}{2}\end{array}$$

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

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

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

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

where Template:Var(Template:Var, Template:Var) is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, Template:Var_Template:Var. Expanding some of the terms in the Hurwitz zeta function gives:

$$H_n=\log (n)+\gamma +\frac{1}{2n}-\frac{1}{12n^2}+\frac{1}{120n^4}-\epsilon ,$$ where 0 < Template:Var < Template:Sfrac.

Template:Mvar can also be expressed as follows where Template:Mvar is the Glaisher–Kinkelin constant:

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

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

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

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 $\ln 2T_k$ in a series that considers the powers of $\frac{1}{T_k}$ (an earlier, less-generalizable proof by Ernesto Cesàro gives the first two terms of the series, with an error term):

$\begin{array}{rl}\gamma & =H_u-\frac{1}{2}\ln 2T_u-\sum _{k=1}^v\frac{R(k)}{T_u^k}-{\mathrm{\Theta }}_v\frac{R(v+1)}{T_u^{v+1}}\end{array}$

From Stirling's approximation follows a similar series:

$\gamma =\ln 2\pi -\sum _{k=2}^{\mathrm{\infty }}\frac{\zeta (k)}{T_k}$

The series of inverse triangular numbers also features in the study of the Basel problemTemplate:Sfn posed by Pietro Mengoli. Mengoli proved that ${\sum }_{k=1}^{\mathrm{\infty }}\frac{1}{2T_k}=1$, a result Jacob Bernoulli later used to estimate the value of $\zeta (2)$, placing it between $1$ and ${\sum }_{k=1}^{\mathrm{\infty }}\frac{2}{2T_k}={\sum }_{k=1}^{\mathrm{\infty }}\frac{1}{T_k}=2$. 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 ${\sum }_{k=1}^{\mathrm{\infty }}\frac{1}{T_k}$:

$\gamma -\ln 2=\ln 2\pi +\sum _{\rho }\frac{2}{\rho }-\sum _{k=1}^{\mathrm{\infty }}\frac{1}{T_k}$

Integrals

Template:Mvar equals the value of a number of definite integrals:

$$\begin{array}{rl}\gamma & =-\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x}\log xdx\\ & =-\underset{0}{\overset{1}{\int }}\log (\log \frac{1}{x})dx\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}(\frac{1}{e^x-1}-\frac{1}{x\cdot e^x})dx\\ & =\underset{0}{\overset{1}{\int }}\frac{1-e^{-x}}{x}dx-\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-x}}{x}dx\\ & =\underset{0}{\overset{1}{\int }}(\frac{1}{\log x}+\frac{1}{1-x})dx\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}(\frac{1}{1+x^k}-e^{-x})\frac{dx}{x},k>0\\ & =2\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-x^2}-e^{-x}}{x}dx,\\ & =\log \frac{\pi }{4}-\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\log x}{{\cosh }^{2}x}dx,\\ & =\underset{0}{\overset{1}{\int }}{H}_{x}dx,\\ & =\frac{1}{2}+\underset{0}{\overset{\mathrm{\infty }}{\int }}\log (1+\frac{\log {(1+\frac{1}{t})}^2}{4{\pi }^2})dt\\ & =1-\underset{0}{\overset{1}{\int }}\{1/x\}dx\\ & =\frac{1}{2}+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{2xdx}{(x^2+1)(e^{2\pi x}-1)}\end{array}$$ where Template:Var_Template:Var 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:

$$\begin{array}{rl}& \underset{0}{\overset{\mathrm{\infty }}{\int }}(\frac{1}{e^x-1}-\frac{1}{x{e}^x})dx=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-x}+x-1}{x[e^x-1]}dx=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{1}{x[e^x-1]}\sum _{m=1}^{\mathrm{\infty }}\frac{(-1{)}^{m+1}{x}^{m+1}}{(m+1)!}dx\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}\sum _{m=1}^{\mathrm{\infty }}\frac{(-1{)}^{m+1}{x}^m}{(m+1)![e^x-1]}dx=\sum _{m=1}^{\mathrm{\infty }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{(-1{)}^{m+1}{x}^m}{(m+1)![e^x-1]}dx=\sum _{m=1}^{\mathrm{\infty }}\frac{(-1{)}^{m+1}}{(m+1)!}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^m}{e^x-1}dx\\ & =\sum _{m=1}^{\mathrm{\infty }}\frac{(-1{)}^{m+1}}{(m+1)!}m!\zeta (m+1)=\sum _{m=1}^{\mathrm{\infty }}\frac{(-1{)}^{m+1}}{m+1}\zeta (m+1)=\sum _{m=1}^{\mathrm{\infty }}\frac{(-1{)}^{m+1}}{m+1}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{m+1}}=\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{m+1}}{m+1}\frac{1}{n^{m+1}}\\ & =\sum _{n=1}^{\mathrm{\infty }}\sum _{m=1}^{\mathrm{\infty }}\frac{(-1{)}^{m+1}}{m+1}\frac{1}{n^{m+1}}=\sum _{n=1}^{\mathrm{\infty }}[\frac{1}{n}-\log (1+\frac{1}{n})]=\gamma \end{array}$$

The integral on the second line of the equation is the definition of the Riemann zeta function, which is Template:Var!Template:Var(Template:Var + 1).

Definite integrals in which Template:Mvar appears include:

$$\begin{array}{rl}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x^2}\log xdx& =-\frac{(\gamma +2\log 2)\sqrt{\pi }}{4}\\ \underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x}{\log }^{2}xdx& ={\gamma }^2+\frac{{\pi }^2}{6}\\ \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-x}\log x}{e^x+1}dx& =\frac{1}{2}{\log }^{2}2-\gamma \end{array}$$

We also have Catalan's 1875 integralTemplate:R

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

One can express Template:Mvar using a special case of Hadjicostas's formula as a double integralTemplate:R with equivalent series:

$$\begin{array}{rl}\gamma & =\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{x-1}{(1-xy)\log xy}dxdy\\ & =\sum _{n=1}^{\mathrm{\infty }}(\frac{1}{n}-\log \frac{n+1}{n}).\end{array}$$

An interesting comparison by SondowTemplate:R is the double integral and alternating series

$$\begin{array}{rl}\log \frac{4}{\pi }& =\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{x-1}{(1+xy)\log xy}dxdy\\ & =\sum _{n=1}^{\mathrm{\infty }}((-1{)}^{n-1}(\frac{1}{n}-\log \frac{n+1}{n})).\end{array}$$

It shows that log Template:Sfrac may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of seriesTemplate:R

$$\begin{array}{rl}\gamma & =\sum _{n=1}^{\mathrm{\infty }}\frac{N_1(n)+N_0(n)}{2n(2n+1)}\\ \log \frac{4}{\pi }& =\sum _{n=1}^{\mathrm{\infty }}\frac{N_1(n)-N_0(n)}{2n(2n+1)},\end{array}$$

where Template:Var₁(Template:Var) and Template:Var₀(Template:Var) are the number of 1s and 0s, respectively, in the base 2 expansion of Template:Mvar.

Series expansions

In general,

$$\gamma =\underset{n\to \mathrm{\infty }}{\lim }(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{n}-\log (n+\alpha ))\equiv \underset{n\to \mathrm{\infty }}{\lim }{\gamma }_n(\alpha )$$

for any Template:Var > −Template:Var. However, the rate of convergence of this expansion depends significantly on Template:Mvar. In particular, Template:Var_Template:Var(1/2) exhibits much more rapid convergence than the conventional expansion Template:Var_Template:Var(0).Template:RTemplate:Sfn 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 Template:Mvar: $$\gamma =\sum _{k=1}^{\mathrm{\infty }}(\frac{1}{k}-\log (1+\frac{1}{k})).$$

The series for Template:Mvar is equivalent to a series Nielsen found in 1897:Template:RTemplate:Sfn

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

In 1910, Vacca found the closely related seriesTemplate:R

$$\begin{array}{rl}\gamma & =\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k\frac{\lfloor {\log }_{2}k\rfloor }{k}\\ & =\frac{1}{2}-\frac{1}{3}+2(\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{7})+3(\frac{1}{8}-\frac{1}{9}+\frac{1}{10}-\frac{1}{11}+\cdots -\frac{1}{15})+\cdots ,\end{array}$$

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

This can be generalized to:

$$\gamma =\sum _{k=1}^{\mathrm{\infty }}\frac{\lfloor {\log }_{B}k\rfloor }{k}\epsilon (k)$$where:$$\epsilon (k)=\{\begin{array}{ll}B-1,& \text{if }B\mid n\\ -1,& \text{if }B\nmid n\end{array}$$

In 1926 Vacca found a second series:

$$\begin{array}{rl}\gamma +\zeta (2)& =\sum _{k=2}^{\mathrm{\infty }}(\frac{1}{{\lfloor \sqrt{k}\rfloor }^2}-\frac{1}{k})\\ & =\sum _{k=2}^{\mathrm{\infty }}\frac{k-{\lfloor \sqrt{k}\rfloor }^2}{k{\lfloor \sqrt{k}\rfloor }^2}\\ & =\frac{1}{2}+\frac{2}{3}+\frac{1}{2^2}\sum _{k=1}^{2\cdot 2}\frac{k}{k+2^2}+\frac{1}{3^2}\sum _{k=1}^{3\cdot 2}\frac{k}{k+3^2}+\cdots \end{array}$$

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

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

Ramanujan, in his lost notebook gave a series that approaches Template:MvarTemplate:R:

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

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

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

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

convergent for Template:Var = 1, 2, ...

A similar series with the Cauchy numbers of the second kind Template:Var_Template:Var isTemplate:R

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

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

$$\gamma =\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{2n}\{{\psi }_n(a)+{\psi }_n(-\frac{a}{1+a}\left)\right\},a>-1$$

where Template:Var_Template:Var(Template:Var) are the Bernoulli polynomials of the second kind, which are defined by the generating function

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

For any rational Template:Mvar this series contains rational terms only. For example, at Template:Var = 1, it becomesTemplate:R

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

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

and

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

where Γ(Template:Var) is the gamma function.Template:R

A series related to the Akiyama–Tanigawa algorithm is

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

where Template:Var_Template:Var(2) are the Gregory coefficients of the second order.Template:R

As a series of prime numbers:

$$\gamma =\underset{n\to \mathrm{\infty }}{\lim }(\log n-\sum _{p\le n}\frac{\log p}{p-1}).$$

Asymptotic expansions

Template:Mvar equals the following asymptotic formulas (where Template:Var_Template:Var is the Template:Mvarth harmonic number):

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

The third formula is also called the Ramanujan expansion.

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

$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}(\log n+\gamma -H_n+\frac{1}{2n})& =\frac{\log (2\pi )-1-\gamma }{2}\\ \sum _{n=1}^{\mathrm{\infty }}(\log \sqrt{n(n+1)}+\gamma -H_n)& =\frac{\log (2\pi )-1}{2}-\gamma \\ \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n(\log n+\gamma -H_n)& =\frac{\log \pi -\gamma }{2}\end{array}$$

Exponential

The constant Template:VarTemplate:I sup is important in number theory. Its numerical value is:Template:R

Template:Block indent Template:VarTemplate:I sup equals the following limit, where Template:Var_Template:Var is the Template:Mvarth prime number:

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

This restates the third of Mertens' theorems.Template:R

We further have the following product involving the three constants Template:Var, Template:Var and Template:Var:

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

Other infinite products relating to Template:VarTemplate:I sup include:

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

These products result from the [[Barnes G-function|Barnes Template:Mvar-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 Template:Mvarth factor is the (Template:Var + 1)th root of

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

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

It also holds thatTemplate:R

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

Published digits

Published decimal expansions of Template:Mvar

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

Generalizations

Stieltjes constants

File:Generalisation of Euler–Mascheroni constant.jpg

Euler's generalized constants abm(Template:Var) for Template:Var > 0.

Euler's generalized constants are given by

$${\gamma }_{\alpha }=\underset{n\to \mathrm{\infty }}{\lim }(\sum _{k=1}^n\frac{1}{k^{\alpha }}-\underset{1}{\overset{n}{\int }}\frac{1}{x^{\alpha }}dx)$$

for 0 < Template:Var < 1, with Template:Mvar as the special case Template:Var = 1.Template:Sfn Extending for Template:Var > 1 gives:

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

with again the limit:

$$\gamma =\underset{a\to 1}{\lim }(\zeta (a)-\frac{1}{a-1})$$

This can be further generalized to

$$c_f=\underset{n\to \mathrm{\infty }}{\lim }(\sum _{k=1}^{n}f(k)-\underset{1}{\overset{n}{\int }}f(x)dx)$$

for some arbitrary decreasing function Template:Mvar. 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:

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

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:Template:R

$$\gamma (a,q)=\underset{x\to \mathrm{\infty }}{\lim }(\sum _{\genfrac{}{}{0ex}{}{0<n\le x}{n\equiv a(\mathrm{mod}q)}}\frac{1}{n}-\frac{\log x}{q}).$$

The basic properties are

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

and if the greatest common divisor gcd(Template:Var,Template:Var) = Template:Var then

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

Masser–Gramain constant

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

$\delta =\underset{n\to \mathrm{\infty }}{\lim }(-\log n+\sum _{k=2}^n\frac{1}{\pi r_k^2})$

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

Further reading

• Derives γ as sums over Riemann zeta functions.
• Julian Havil (2003): GAMMA: Exploring Euler's Constant, Princeton University Press, ISBN 978-0-69114133-6.
• with an Appendix by Sergey Zlobin

External links

• Template:Springer
• Jonathan Sondow.
• Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
• Further formulae which make use of the constant: Gourdon and Sebah (2004).