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Euler–Mascheroni constant

A mathematical constant denoted γ that links harmonic numbers and the natural logarithm; appears in analysis, special functions and number theory with several integral and series representations.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: May 18, 2026

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Overview

The Euler–Mascheroni constant, commonly written as γ, is a real constant that arises in analysis and number theory. It measures the limiting difference between the harmonic numbers and the natural logarithm and appears in many identities, integrals and limits. Historical accounts attribute the first systematic study of the quantity to Leonhard Euler and later computations and interest to Lorenzo Mascheroni. General background on the mathematical areas where the constant appears can be found in texts on analysis and number theory; biographical and historical treatments discuss the contributions of Euler and Mascheroni. $\gamma$

Definition and common representations

The constant is defined as the limit as n grows without bound of the difference between the nth harmonic number and the natural logarithm of n. In words: γ = lim as n → infinity of (H_n − ln n), where H_n = 1 + 1/2 + 1/3 + ... + 1/n. This definition immediately explains the role of γ as a correction term when replacing discrete harmonic sums by the continuous logarithm.

There are many equivalent series and integral representations used in analysis and computation. Common forms expressed in elementary terms include:

a series form: γ = sum_{k=1}^∞ (1/k − ln(1 + 1/k));
an integral related to the Gamma function: γ = − integral from 0 to ∞ of e^{−t} ln t dt, which follows from differentiating the Gamma function at 1;
a floor integral: γ = integral from 1 to ∞ of (1/floor(t) − 1/t) dt, emphasizing the discrete-to-continuous comparison.

Other convenient expressions can be derived for numerical work and theoretical analysis. Standard expositions collecting these forms appear in references on series and special functions as well as treatments specifically devoted to the Gamma function and related topics. $\gamma$

Relations with special functions

The Euler–Mascheroni constant is tightly connected with the Gamma and digamma functions. If Γ(x) denotes the Gamma function, its logarithmic derivative ψ(x) = Γ'(x)/Γ(x) is the digamma function. The value at x = 1 gives ψ(1) = −γ, so γ equals −ψ(1). This relation places γ among the special values of classical functions that are central in analytic continuation and factorial generalization. More technical accounts of these relationships are available in sources on the digamma function and detailed references on the Gamma function and its derivatives. $0.5772156649$

Numeric value and computation

The constant begins approximately 0.5772156649... and is commonly rounded to several decimal places in elementary references. Historically, Mascheroni attempted decimal approximations and published results; some early hand computations contained errors in a few digits, which later computational work corrected. Modern computation has extended γ to very large numbers of digits using high-precision arithmetic and rapidly convergent series. For accessible discussions of algorithms and records of computed digits see sources on numerical computation and historical computational notes. $\gamma =\lim _{t\to \infty }\left(\sum _{n=1}^{t}{\frac {1}{n}}-\log(t)\right)$

Arithmetic nature and open questions

Despite intensive study, the exact arithmetic nature of γ remains an open problem in mathematics. It is not known whether γ is rational or irrational, nor whether it is algebraic or transcendental. Several conditional results and partial theorems relate possible algebraic relations of γ to deep conjectures in transcendence theory and the arithmetic of special values of L-functions. Surveys and research-level discussions describe the state of knowledge and the principal lines of attack on these questions. Readers seeking an overview of the open problems and known partial results may consult research surveys and specialist articles. Surveys on the arithmetic status and research summaries provide further context. $\gamma =\int _{1}^{\infty }\left({\frac {1}{\lfloor t\rfloor }}-{\frac {1}{t}}\right)\mathrm {d} t$

Euler–Mascheroni appears as the leading constant in asymptotic expansions that compare discrete sums to integrals. For example, the harmonic numbers have the expansion H_n = ln n + γ + 1/(2n) − 1/(12 n^2) + O(1/n^4) as n → infinity, an expansion useful in numerical estimates and analytic proofs. More generally, γ is the zeroth Stieltjes constant, usually denoted γ_0, which arises as the constant term in the Laurent expansion of the Riemann zeta function around s = 1. The family of Stieltjes constants γ_n generalizes γ and appears in more refined expansions. Texts on asymptotic methods and zeta-function theory discuss these connections at greater length. Asymptotic expansions and Stieltjes constants are standard topics in such treatments. $\mathrm {\Psi } _{0}(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\log(\Gamma (x))={\frac {\Gamma '(x)}{\Gamma (x)}}$

Applications and examples

Euler–Mascheroni arises in diverse contexts: in evaluations of certain definite integrals that involve logarithms, in limits of combinatorial sums, in expansions of special functions, and in correction terms when approximating sums by integrals. It also appears in analytic number theory through special functions that model prime-counting behavior and related estimates. Practical examples include limits that convert sums to integrals with an explicit constant term, formulae for digamma values at rational points, and corrections to approximation formulas in probability and statistics. For expository examples and numerical tables, see introductory articles and computational guides on special functions and constants. Expository treatments and numerical guides give numerous worked examples and applications. $x=1$ $\mathrm {\Psi } _{0}(1)=-\gamma$ $\gamma$

Summary: The Euler–Mascheroni constant γ ≈ 0.5772156649 is a fundamental constant linking discrete harmonic sums and the continuous logarithm. It appears throughout analysis and number theory, is deeply connected with the Gamma and digamma functions, and remains the subject of open problems about its arithmetic nature. The constant's many representations make it a useful and recurring quantity in both theoretical work and numerical computation.

The blue area represents Euler's constant.

General

Despite great efforts, it is still unknown whether this number is rational or irrational, whether it is algebraic or transcendental. However, it is strongly suspected that it is at least an irrational number. The first concrete attempt to prove this was made in 1926 by Paul Émile Appell with the help of Joseph Ser's development mentioned below. By calculating the continued fraction development of γ $\gamma$ (sequence A002852 in OEIS)

$\gamma =\left[0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,3,7,1,7,1,1,5,1,49,4,1,65,1,4,7,11,1,399,2,1,3,2,1,2,1,5,3,2,1,\dotsc \right]$

lower bounds are obtained for positive integers and with γ $\gamma ={\tfrac {p}{q}}$ (for example, 475,006 denominators give the estimate $q>10^{244.663}$ ).

In contrast to square roots of rational numbers in the Pythagorean theorem and to the circle number π $\pi$ in the circumference and area of a circle with rational radius, Euler's constant does not occur in finite elementary geometric problems. However, there are many engineering problems that lead to the summation of the finite harmonic series $H_{n}$ , such as the centre of gravity problem of the cantilever or the problem of the optimal elevation of rows of seats in theatres and cinemas. Euler's constant appears in many problems in calculus, number theory and function theory and especially in special functions.

Convergence

The existence of Euler's constant results from the telescope sum

$\gamma =\lim _{n\rightarrow \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n+1)\right)=\sum _{k=1}^{\infty }\left({\frac {1}{k}}-\ln {\frac {k+1}{k}}\right).$

Since $\ln(n+1)-\ln(n)$ is a zero sequence, can be $\ln(n)$ used instead of $\ln(n+1)$ the defining limit. The following applies

${\frac {1}{k}}-\ln {\frac {k+1}{k}}={\frac {1}{k}}-\int _{k}^{k+1}{\frac {\mathrm {d} x}{x}}=\int _{k}^{k+1}{\frac {x-k}{xk}}\,\mathrm {d} x={\frac {1}{k}}\int _{0}^{1}{\frac {x}{x+k}}\,\mathrm {d} x.$

About

${\frac {1}{2(k+1)}}=\int _{0}^{1}{\frac {x}{1+k}}\,\mathrm {d} x\leq \int _{0}^{1}{\frac {x}{x+k}}\,\mathrm {d} x\leq \int _{0}^{1}{\frac {x}{k}}\,\mathrm {d} x={\frac {1}{2k}}$

therefore applies

${\frac {1}{2k\cdot (k+1)}}\leq {\frac {1}{k}}-\ln {\frac {k+1}{k}}\leq {\frac {1}{2k^{2}}}$

and thus the sum converges according to the major criterion.

In particular, it follows from this elementary argument and

$\sum _{k=1}^{\infty }{\frac {1}{k\cdot (k+1)}}=\sum _{k=1}^{\infty }\left({\frac {1}{k}}-{\frac {1}{k+1}}\right)=1$

and the Basel problem that

${\frac {1}{2}}\leq \gamma \leq {\frac {\pi ^{2}}{12}}$

applies.

The Euler-Mascheroni constant in mathematical problems

Euler's constant occurs frequently in mathematics and sometimes quite unexpectedly in various sub-areas. It mainly occurs in limit value processes of number sequences and functions as well as in limit values of differential and integral calculus. The occurrence can be subdivided (as with other mathematical constants) according to the type of limit value:

1. as a function value or limit value of special functions.

The value γ $\gamma$ is the negative of the derivative of the gamma function at the point 1, i.e.

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

This gives the following limit representations, where $\zeta (s)$ denotes the Riemann zeta function and ψ $\psi (z)$ denotes the digamma function:

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

$\lim _{z\to 0}\left\{\Gamma (z)-{\frac {1}{z}}\right\}=\lim _{z\to 0}\left\{\psi (z)+{\frac {1}{z}}\right\}=-\gamma$

$\lim _{z\to 0}{\frac {1}{z}}\left\{{\frac {1}{\Gamma (1+z)}}-{\frac {1}{\Gamma (1-z)}}\right\}=2\gamma$

$\lim _{z\to 0}{\frac {1}{z}}\left\{{\frac {1}{\psi (1-z)}}-{\frac {1}{\psi (1+z)}}\right\}={\frac {\pi ^{2}}{3\gamma ^{2}}}$

2. in the development of special functions, e.g. in the series development of the integral logarithm of Leopold Schendel, the Bessel functions or the Weierstrass representation of the gamma function.

3. in the evaluation of certain integrals.

There is a rich abundance here, for example:

${\begin{aligned}\gamma &=-\int _{0}^{1}\ln(-\ln x)\,\mathrm {d} x\\\gamma &=-\int _{0}^{\infty }e^{-x}\ln x\,\mathrm {d} x\\\gamma &=\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{xe^{x}}}\right)\,\mathrm {d} x\\\gamma &=\int _{0}^{1}\left({\frac {1}{\ln x}}+{\frac {1}{1-x}}\right)\,\mathrm {d} x\\\gamma &={\frac {1}{2}}+2\int _{0}^{\infty }{\frac {\sin(\arctan x)}{(e^{2\pi x}-1){\sqrt {1+x^{2}}}}}\,\mathrm {d} x\end{aligned}}$

or also

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

There are also many invariant parameter integrals, e.g.:

${\begin{aligned}\gamma &=\int _{0}^{\infty }\left({\frac {1}{x^{k}+1}}-e^{-x}\right){\frac {\mathrm {d} x}{x}},\quad k>0\\\gamma &=\int _{0}^{\infty }\left({\frac {1}{kx+1}}-e^{-kx}\right){\frac {\mathrm {d} x}{x}},\quad k>0\end{aligned}}$

One can $\gamma$ also express γ as a double integral (J. Sondow 2003, 2005) with the equivalent series:

$\gamma =\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-xy)\ln(xy)}}\,\mathrm {d} x\,\mathrm {d} y=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right)$ .

There is an interesting comparison (J. Sondow 2005) of the double integral and the alternating series:

$\ln \left({\frac {4}{\pi }}\right)=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1+xy)\ln(xy)}}\,\mathrm {d} x\,\mathrm {d} y=\sum _{n=1}^{\infty }(-1)^{n-1}\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right)$ .

In this sense, we can say that $\ln {\big (}{\tfrac {4}{\pi }}{\big )}$ is the "alternating Euler's constant" (sequence A094640 in OEIS).

Furthermore, these two constants are linked to the pair

$\sum _{n=1}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}=\gamma ,$

$\sum _{n=1}^{\infty }{\frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}}=\ln \left({\frac {4}{\pi }}\right)$

of series, where $N_{1}(n)$ and are $N_{0}(n)$ the number of ones and zeros, respectively, in the binary expansion of (Sondow 2010).

Furthermore, there is an equally rich abundance of infinite sums and products, such as

${\begin{aligned}e^{\gamma }&=\lim _{n\to \infty }{\frac {1}{\ln n}}\prod _{p\leq n,p{\text{ prim}}}\left(1-{\frac {1}{p}}\right)^{-1}\\{\frac {6}{\pi ^{2}}}e^{\gamma }&=\lim _{n\to \infty }{\frac {1}{\ln n}}\prod _{p\leq n,p{\text{ prim}}}\left(1+{\frac {1}{p}}\right)\\\gamma &=\lim _{x\to 1^{+}}\sum _{n=1}^{\infty }\left({\frac {1}{n^{x}}}-{\frac {1}{x^{n}}}\right).\end{aligned}}$

4. as the limit value of series. The simplest example results from the limit value definition:

$\gamma =\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right)$ .

Series with rational terms come from Euler, Fontana and Mascheroni, Giovanni Enrico Eugenio Vacca, S. Ramanujan and Joseph Ser. There are countless variations on series with irrational members whose members consist of rationally weighted values of the Riemann zeta function at the odd argument positions ζ(3), ζ(5), .... An example of a particularly fast converging series is:

$\sum _{n=1}^{\infty }{\frac {\zeta (2n+1)-1}{(2n+1)2^{2n}}}=1+\ln 2-\ln 3-\gamma =$ 0.0173192269903...

Another series results from the Kummer series of the gamma function:

$\gamma =\ln \pi -4\ln \Gamma {\big (}{\tfrac {3}{4}}{\big )}+{\frac {4}{\pi }}\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {\ln(2k+1)}{2k+1}}$

Designations

One can say that Euler's constant is the constant with the most designations. Euler himself designated it with C and occasionally with O or n. However, it is doubtful whether he intended to introduce an independent symbol for his constant. Mascheroni did not designate the constant with γ - as is often claimed - but with A. The γ-misunderstanding stems from the article by J. W. L. Glaisher, which is often quoted without verification (although Glaisher explicitly notes there that he has not seen Mascheroni's book):

"Euler's constant (which throughout this note will be called γ after Mascheroni, De Morgan, &c.) [...]It
is clearly convenient that the constant should generally be denoted by the same letter. Euler used C and O for it; Legendre, Lindman, &c., C; De Haan A; and Mascheroni, De Morgan, Boole, &c., have written it γ, which is clearly the most suitable, if it is to have a distinctive letter assigned to it. It has sometimes (as in Crelle, t. 57, p. 128) been quoted as Mascheroni's constant, but it is evident that Euler's labours have abundantly justified his claim to its being named after him."

- J. W. L. Glaisher: On the history of Euler's constant, 1872, p. 25 and 30

Other mathematicians use the designations C, c, ℭ, H, γ, E, K, M, l. The origin of the designation γ used today is not certain. Carl Anton Bretschneider used the designation γ alongside c in an article written in 1835 and published in 1837, Augustus De Morgan introduced the designation γ in a textbook published in parts from 1836 to 1842 as part of his treatment of the gamma function.

Generalisations

Euler's constant knows several generalisations. The most important and best known is that of the Stieltjes constant:

$\gamma _{n}:=\lim _{N\to \infty }\left(\sum _{k=1}^{N}{\frac {\log ^{n}k}{k}}-{\frac {\log ^{n+1}N}{n+1}}\right),\quad n=0,1,2,\dotsc$

Number of calculated decimal places

In 1734, Leonhard Euler calculated six decimal places (five valid), later 16 places (15 valid). In 1790, Lorenzo Mascheroni calculated 32 decimal places (30 valid ones), of which, however, the three places 20 to 22 are wrong - apparently due to a clerical error, but they are given several times in the book. The error was the cause of several recalculations.

Number of published valid decimal places of γ
Date	Jobs	Author
1734	5	Leonhard Euler
1735	15	Leonhard Euler
1790	19	Lorenzo Mascheroni
1809	22	Johann Georg Soldner
1811	22	Carl Friedrich Gauss
1812	40	Friedrich Bernhard Gottfried Nicolai
1826	19	Adrien-Marie Legendre
1857	34	Christian Fredrik Lindman
1861	41	Ludwig Oettinger
1867	49	William Shanks
1871	99	J. W. L. Glaisher
1871	101	William Shanks
1877	262	John Couch Adams
1952	328	John William Wrench, Jr.
1961	1.050	Helmut Fischer & Karl Zeller
1962	1.270	Donald E. Knuth
1962	3.566	Dura W. Sweeney
1973	4.879	William A. Beyer & Michael S. Waterman
1976	20.700	Richard P. Brent
1979	30.100	Richard P. Brent & Edwin M. McMillan
1993	172.000	Jonathan Borwein
1997	1.000.000	Thomas Papanikolaou
1998	7.286.255	Xavier Gourdon
1999	108.000.000	Xavier Gourdon & Patrick Demichel
8 December 2006	116.580.041	Alexander J. Yee & Raymond Chan
18 January 2009	14.922.244.771	Alexander J. Yee & Raymond Chan
13 March 2009	29.844.489.545	Alexander J. Yee & Raymond Chan
22 December 2013	119.377.958.182	Alexander J. Yee
15 March 2016	160.000.000.000	Peter Trueb
18 May 2016	250.000.000.000	Ron Watkins
23 August 2017	477.511.832.674	Ron Watkins
26 May 2020	600.000.000.100	Seungmin Kim & Ian Cutress