Euler's constant#Euler-constant function

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".{{sfn|Lagarias|2013}} Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations {{math|C}} and {{math|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 {{math|A}} and {{math|a}} for the constant. Other computations were done by Johann von Soldner in 1809, who used the notation {{math|H}}. The notation {{math|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.{{sfn|Lagarias|2013}} For example, the German mathematician Carl Anton Bretschneider used the notation {{math|γ}} in 1835,{{sfn|Bretschneider|1837|loc="{{math|1=γ = c}} = {{gaps|0,577215|664901|532860|618112|090082|3...}}" on [https://books.google.com/books?hl=fi&id=OAoPAAAAIAAJ&pg=PA260 p. 260]}} and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.{{r|DeMorgan183642}} Euler's constant was also studied by the Indian mathematician Srinivasa Ramanujan who published one paper on it in 1917.{{Cite book |last=Brent |first=Richard P. |date=1994 |chapter=Ramanujan and Euler's Constant |title=Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics |series=Proceedings of Symposia in Applied Mathematics |chapter-url=https://maths-people.anu.edu.au/~brent/pd/Euler_CARMA_10.pdf |volume=48 |pages=541–545|doi=10.1090/psapm/048/1314887 |isbn=978-0-8218-0291-5 }} David Hilbert mentioned the irrationality of {{math|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.

Euler's constant appears frequently in mathematics, especially in number theory and analysis.{{Cite web |last=Sondow |first=Jonathan |date=2004 |title=The Euler constant: γ |url=http://numbers.computation.free.fr/Constants/Gamma/gamma.html |access-date=2024-11-01}} Examples include, among others, the following places: (where '*' means that this entry contains an explicit equation):

• The Weierstrass product formula for the gamma function and the Barnes G-function.{{cite journal |last=Davis |first=P. J. |date=1959 |title=Leonhard Euler's Integral: A Historical Profile of the Gamma Function |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3104 |url-status=dead |journal=American Mathematical Monthly |volume=66 |issue=10 |pages=849–869 |doi=10.2307/2309786 |jstor=2309786 |archive-url=https://web.archive.org/web/20121107190256/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3104 |archive-date=7 November 2012 |access-date=3 December 2016}}{{Cite web |title=DLMF: §5.17 Barnes' 𝐺-Function (Double Gamma Function) ‣ Properties ‣ Chapter 5 Gamma Function |url=https://dlmf.nist.gov/5.17 |access-date=2024-11-01 |website=dlmf.nist.gov}}
• The asymptotic expansion of the gamma function, $\backslash Gamma(1/x)\backslash sim\; x-\backslash gamma$.
• Evaluations of the digamma function at rational values.{{Cite web |last=Weisstein |first=Eric W. |title=Digamma Function |url=https://mathworld.wolfram.com/DigammaFunction.html |access-date=2024-10-30 |website=mathworld.wolfram.com |language=en}}
• The Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants.{{Cite web |last=Weisstein |first=Eric W. |title=Stieltjes Constants |url=https://mathworld.wolfram.com/StieltjesConstants.html |access-date=2024-11-01 |website=mathworld.wolfram.com |language=en}}
• Values of the derivative of the Riemann zeta function and Dirichlet beta function.{{rp|137}}
• In connection to the Laplace and Mellin transform.{{Cite book |last=Williams |first=John |title=Laplace transforms |date=1973 |publisher=Allen & Unwin |isbn=978-0-04-512021-5 |series=Problem solvers |location=London}}{{Cite web |title=DLMF: §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations |url=https://dlmf.nist.gov/2.5 |access-date=2024-11-01 |website=dlmf.nist.gov}}
• In the regularization/renormalization of the harmonic series as a finite value.
• Expressions involving the exponential and logarithmic integral.*{{Cite web |title=DLMF: §6.6 Power Series ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals |url=https://dlmf.nist.gov/6.6 |access-date=2024-11-01 |website=dlmf.nist.gov}}{{Cite web |last=Weisstein |first=Eric W. |title=Logarithmic Integral |url=https://mathworld.wolfram.com/LogarithmicIntegral.html |access-date=2024-11-01 |website=mathworld.wolfram.com |language=en}}
• A definition of the cosine integral.*
• In relation to Bessel functions.{{Cite web |title=DLMF: §10.32 Integral Representations ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions |url=https://dlmf.nist.gov/10.32 |access-date=2024-11-01 |website=dlmf.nist.gov}}{{Cite web |title=DLMF: §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions |url=https://dlmf.nist.gov/10.22 |access-date=2024-11-01 |website=dlmf.nist.gov}}{{Cite web |title=DLMF: §10.8 Power Series ‣ Bessel Functions and Hankel Functions ‣ Chapter 10 Bessel Functions |url=https://dlmf.nist.gov/10.8 |access-date=2024-11-01 |website=dlmf.nist.gov}}{{Cite web |title=DLMF: §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions |url=https://dlmf.nist.gov/10.24 |access-date=2024-11-01 |website=dlmf.nist.gov}}
• Asymptotic expansions of modified Struve functions.{{Cite web |title=DLMF: §11.6 Asymptotic Expansions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions |url=https://dlmf.nist.gov/11.6 |access-date=2024-11-01 |website=dlmf.nist.gov}}
• In relation to other special functions.{{Cite web |title=DLMF: §13.2 Definitions and Basic Properties ‣ Kummer Functions ‣ Chapter 11 Confluent Hypergeometric Functions |url=https://dlmf.nist.gov/13.2 |access-date=2024-11-01 |website=dlmf.nist.gov}}{{Cite web |title=DLMF: §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions |url=https://dlmf.nist.gov/9.12 |access-date=2024-11-01 |website=dlmf.nist.gov}}

• An inequality for Euler's totient function.{{Cite journal |last1=Rosser |first1=J. Barkley |last2=Schoenfeld |first2=Lowell |date=1962 |title=Approximate formulas for some functions of prime numbers |url=https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-6/issue-1/Approximate-formulas-for-some-functions-of-prime-numbers/10.1215/ijm/1255631807.full |journal=Illinois Journal of Mathematics |volume=6 |issue=1 |pages=64–94 |doi=10.1215/ijm/1255631807 |issn=0019-2082}}
• The growth rate of the divisor function.{{Cite book |last1=Hardy |first1=Godfrey H. |title=An introduction to the theory of numbers |last2=Wright |first2=Edward M. |last3=Silverman |first3=Joseph H. |date=2008 |publisher=Oxford University Press |isbn=978-0-19-921986-5 |editor-last=Heath-Brown |editor-first=D. R. |edition=6th |series=Oxford mathematics |location=Oxford New York Auckland |page=469-471}}
• A formulation of the Riemann hypothesis.{{Cite journal |last=Robin |first=Guy |date=1984 |title=Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann |url=http://zakuski.utsa.edu/~jagy/Robin_1984.pdf |journal=Journal de mathématiques pures et appliquées |volume=63 |pages=187–213}}
• The third of Mertens' theorems.*
• The calculation of the Meissel–Mertens constant.{{Cite web |last=Weisstein |first=Eric W. |title=Mertens Constant |url=https://mathworld.wolfram.com/MertensConstant.html |access-date=2024-11-01 |website=mathworld.wolfram.com |language=en}}
• Lower bounds to specific prime gaps.{{Cite journal |last=Pintz |first=János |date=1997-04-01 |title=Very Large Gaps between Consecutive Primes |url=https://www.sciencedirect.com/science/article/pii/S0022314X97920813 |journal=Journal of Number Theory |volume=63 |issue=2 |pages=286–301 |doi=10.1006/jnth.1997.2081 |issn=0022-314X}}
• 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.{{Cite web |title=Heuristics: Deriving the Wagstaff Mersenne Conjecture |url=https://t5k.org/mersenne/heuristic.html |access-date=2024-11-01 |website=t5k.org}}
• An estimation of the efficiency of the euclidean algorithm.{{Cite web |last=Weisstein |first=Eric W. |title=Porter's Constant |url=https://mathworld.wolfram.com/PortersConstant.html |access-date=2024-11-01 |website=mathworld.wolfram.com |language=en}}
• Sums involving the Möbius and von Mangolt function.
• Estimate of the divisor summatory function of the Dirichlet hyperbola method.{{Cite book |last=Tenenbaum |first=Gérald |url=https://books.google.com/books?id=UEk-CgAAQBAJ&dq=dirichlet+hyperbola+method&pg=PA360 |title=Introduction to Analytic and Probabilistic Number Theory |date=2015-07-16 |publisher=American Mathematical Soc. |isbn=978-0-8218-9854-3 |language=en}}

• 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.{{r|CavesFuchs1996}}
• 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.{{r|ConnallonHodgins2021}}

The number {{math|γ}} has not been proved algebraic or transcendental. In fact, it is not even known whether {{math|γ}} is irrational. The ubiquity of {{math|γ}} revealed by the large number of equations below and the fact that {{math|{{var|γ}}}} has been called the third most important mathematical constant after Pi and E (mathematical constant){{Cite web |title=Eulers Constant |url=https://num.math.uni-goettingen.de/~skraemer/gamma.html |access-date=2024-10-19 |website=num.math.uni-goettingen.de}}{{Cite book |last=Finch |first=Steven R. |url=https://books.google.com/books?id=Pl5I2ZSI6uAC |title=Mathematical Constants |date=2003-08-18 |publisher=Cambridge University Press |isbn=978-0-521-81805-6 |language=en}} makes the irrationality of {{math|γ}} a major open question in mathematics.{{Cite web |last=Waldschmidt |first=Michel |date=2023 |title=Some of the most famous open problems in number theory |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/OpenPbsNT.pdf}}{{r|Sondow2003a}}{{Cite book |last1=Conway |first1=John H. |url=https://books.google.com/books?id=0--3rcO7dMYC |title=The Book of Numbers |last2=Guy |first2=Richard |date=1998-03-16 |publisher=Springer Science & Business Media |isbn=978-0-387-97993-9 |language=en}}

{{unsolved|mathematics|Is Euler's constant irrational? If so, is it transcendental?}}

However, some progress has been made. In 1959 Andrei Shidlovsky proved that at least one of Euler's constant {{math|γ}} and the Gompertz constant {{math|δ}} is irrational;{{r|Aptekarev2009}}{{Cite web |last=Waldschmidt |first=Michel |date=2023 |title=On Euler's Constant |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/EulerConstant.pdf |place=Sorbonne Université, Institut de Mathématiques de Jussieu, Paris}} Tanguy Rivoal proved in 2012 that at least one of them is transcendental.{{r|Rivoal2012}} Kurt Mahler showed in 1968 that the number $\backslash frac\; \backslash pi\; 2\backslash frac\{Y\_0(2)\}\{J\_0(2)\}-\backslash gamma$ is transcendental, where $J\_0$ and $Y\_0$ are the usual Bessel functions.{{r|Mahler1968}}{{sfn|Lagarias|2013}} It is known that the transcendence degree of the field $\backslash mathbb\; Q(e,\backslash gamma,\backslash delta)$ is at least two.{{sfn|Lagarias|2013}}

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

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

is algebraic, if {{math|q ≥ 2}} and {{math|1 ≤ a < q}}; this family includes the special case {{math|1=γ(2,4) = γ/4}}.{{sfn|Lagarias|2013}}{{r|RamMurtySaradha2010}}

Using the same approach, in 2013, M. Ram Murty and A. Zaytseva showed that the generalized Euler constants have the same property,

{{sfn|Lagarias|2013}}{{r|MurtyZaytseva2013}}

{{Cite journal |last1=Diamond |first1=H. G. |last2=Ford |first2=K. |title=Generalized Euler constants |journal=Mathematical Proceedings of the Cambridge Philosophical Society |date=2008 |volume=145 |issue=1 |pages=27–41 |publisher=Cambridge University Press |doi=10.1017/S0305004108001187 |arxiv=math/0703508|bibcode=2008MPCPS.145...27D }} where the generalized Euler constant are defined as

$$\backslash gamma(\backslash Omega)\; =\; \backslash lim\_\{x\backslash rightarrow\backslash infty\}\; \backslash left(\; \backslash sum\_\{n=1\}^x\; \backslash frac\{1\_\backslash Omega(n)\}\{n\}\; -\; \backslash log\; x\; \backslash cdot\; \backslash lim\_\{x\backslash rightarrow\backslash infty\}\; \backslash frac\{\; \backslash sum\_\{n=1\}^x\; 1\_\backslash Omega\; (n)\; \}\{x\}\; \backslash right),$$

where {{tmath|\Omega}} is a fixed list of prime numbers, $1\_\backslash Omega(n)\; =0$ if at least one of the primes in {{tmath|\Omega}} is a prime factor of {{tmath|n}}, and $1\_\backslash Omega(n)\; =1$ otherwise. In particular, {{tmath|1=\gamma(\empty)=\gamma}}.

Using a continued fraction analysis, Papanikolaou showed in 1997 that if {{math|γ}} is rational, its denominator must be greater than 10²⁴⁴⁶⁶³.{{r|HaiblePapanikolaou1998|Papanikolaou1997}} If {{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} is a rational number, then its denominator must be greater than 10¹⁵⁰⁰⁰.{{sfn|Lagarias|2013}}

Euler's constant is conjectured not to be an algebraic period,{{sfn|Lagarias|2013}} but the values of its first 10⁹ decimal digits seem to indicate that it could be a normal number.{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant Digits |url=https://mathworld.wolfram.com/Euler-MascheroniConstantDigits.html |access-date=2024-10-19 |website=mathworld.wolfram.com |language=en}}

The simple continued fraction expansion of Euler's constant is given by:{{r|OEIS_A002852}}

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

which has no apparent pattern. It is known to have at least 16,695,000,000 terms,{{r|OEIS_A002852}} and it has infinitely many terms if and only if {{mvar|γ}} is irrational.

File:KhinchinBeispiele.svg

Numerical evidence suggests that both Euler's constant {{math|{{var|γ}}}} as well as the constant {{math|{{var|e}}{{i sup|1px|{{var|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 $\backslash lim\_\{n\backslash to\backslash infty\}q\_n^\{1/n\}$ appears to converge to Lévy's constant in both cases.{{Cite journal |last=Brent |first=Richard P. |date=1977 |title=Computation of the Regular Continued Fraction for Euler's Constant |url=https://www.jstor.org/stable/2006010 |journal=Mathematics of Computation |volume=31 |issue=139 |pages=771–777 |doi=10.2307/2006010 |jstor=2006010 |issn=0025-5718}} However neither of these limits has been proven.{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant Continued Fraction |url=https://mathworld.wolfram.com/Euler-MascheroniConstantContinuedFraction.html |access-date=2024-09-23 |website=mathworld.wolfram.com |language=en}}

There also exists a generalized continued fraction for Euler's constant.{{Citation |last1=Pilehrood |first1=Khodabakhsh Hessami |title=On a continued fraction expansion for Euler's constant |date=2013-12-29 |last2=Pilehrood |first2=Tatiana Hessami|arxiv=1010.1420 }}

A good simple approximation of {{math|{{var|γ}}}} is given by the reciprocal of the square root of 3 or about 0.57735:{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant Approximations |url=https://mathworld.wolfram.com/Euler-MascheroniConstantApproximations.html |access-date=2024-10-19 |website=mathworld.wolfram.com |language=en}}

:$\backslash frac1\backslash sqrt\; \{3\}=0+\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{2+\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{2+\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{2+\backslash dots\}\}\}\}\}\}\}$

with the difference being about 1 in 7,429.

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

$$-\backslash gamma\; =\; \backslash Gamma\text{'}(1)\; =\; \backslash Psi(1).$$

This is equal to the limits:

Further limit results are:{{r|Krämer2005}}

\lim_{z\to 0}\frac1{z}\left(\frac1{\Psi(1-z)} - \frac1{\Psi(1+z)}\right) &= \frac{\pi^2}{3\gamma^2}. \end{align}

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

&= \lim\limits_{m\to\infty}\sum_{k=1}^m{m \choose k}\frac{(-1)^k}{k}\log\big(\Gamma(k+1)\big). \end{align}

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

&= \log\frac4{\pi} + \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{2^{m-1}m}.\end{align}

The constant $\backslash gamma$ can also be expressed in terms of the sum of the reciprocals of non-trivial zeros $\backslash rho$ of the zeta function:{{Cite arXiv

| last = Wolf

| first = Marek

| title = 6+infinity new expressions for the Euler-Mascheroni constant

| year = 2019

| eprint = 1904.09855

| class = math.NT

| quote = "The above sum is real and convergent when zeros $\backslash rho$ and complex conjugate $\backslash bar\{\backslash rho\}$ are paired together and summed according to increasing absolute values of the imaginary parts of {{nowrap|$\backslash rho$.}}"}} See formula 11 on page 3. Note the typographical error in the numerator of Wolf's sum over zeros, which should be 2 rather than 1.

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

Other series related to the zeta function include:

&= \lim_{n\to\infty}\left(\frac{2n-1}{2n} - \log n + \sum_{k=2}^n \left(\frac1{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 \frac1{t+1} - n \log 2+ O \left (\frac1{2^{n}\, e^{2^n}}\right)\right).\end{align}

The error term in the last equation is a rapidly decreasing function of {{mvar|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:{{r|Sondow1998}}

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

$$\backslash gamma\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash frac1\{n\}\backslash ,\; \backslash sum\_\{k=1\}^n\; \backslash left(\backslash left\backslash lceil\; \backslash frac\{n\}\{k\}\; \backslash right\backslash rceil\; -\; \backslash frac\{n\}\{k\}\backslash right)$$

where {{math|{{ceil| }}}} are ceiling brackets.

This formula indicates that when taking any positive integer {{mvar|n}} and dividing it by each positive integer {{mvar|k}} less than {{mvar|n}}, the average fraction by which the quotient {{math|{{var|n}}/{{var|k}}}} falls short of the next integer tends to {{mvar|γ}} (rather than 0.5) as {{mvar|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:

$$\backslash gamma\; =\backslash lim\_\{n\backslash to\backslash infty\}\backslash left(\; \backslash sum\_\{k=1\}^n\; \backslash frac1\{k\}\; -\; \backslash log\; n\; -\backslash sum\_\{m=2\}^\backslash infty\; \backslash frac\{\backslash zeta(m,n+1)\}\{m\}\backslash right),$$

where {{math|{{var|ζ}}({{var|s}}, {{var|k}})}} is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, {{math|{{var|H}}{{sub|{{var|n}}}}}}. Expanding some of the terms in the Hurwitz zeta function gives:

$$H\_n\; =\; \backslash log(n)\; +\; \backslash gamma\; +\; \backslash frac1\{2n\}\; -\; \backslash frac1\{12n^2\}\; +\; \backslash frac1\{120n^4\}\; -\; \backslash varepsilon,$$

where {{math|0 < {{var|ε}} < {{sfrac|1|252{{var|n}}{{sup|6}}}}.}}

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

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

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

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

Numerous formulations have been derived that express $\backslash gamma$ in terms of sums and logarithms of triangular numbers.{{Cite journal

| last = Boya

| first = L.J.

| title = Another relation between π, e, γ and ζ(n)

| journal = Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

| volume = 102

| pages = 199–202

| year = 2008

| issue = 2

| url = https://doi.org/10.1007/BF03191819

| doi = 10.1007/BF03191819

| bibcode = 2008RvMad.102..199B

| quote = "γ/2 in (10) reflects the residual (finite part) of ζ(1)/2, of course."

}} See formulas 1 and 10.{{Cite journal

| last = Sondow

| first = Jonathan

| title = Double Integrals for Euler's Constant and $\backslash textstyle\; \backslash frac\{4\}\{\backslash pi\}$ and an Analog of Hadjicostas's Formula

| journal = The American Mathematical Monthly

| volume = 112

| issue = 1

| year = 2005

| pages = 61–65

| url = https://doi.org/10.2307/30037385

| doi = 10.2307/30037385

| jstor = 30037385

| access-date = 2024-04-27

}}{{Cite journal

| last = Chen

| first = Chao-Ping

| title = Ramanujan's formula for the harmonic number

| journal = Applied Mathematics and Computation

| volume = 317

| year = 2018

| pages = 121–128

| issn = 0096-3003

| doi = 10.1016/j.amc.2017.08.053

| url = https://www.sciencedirect.com/science/article/pii/S0096300317306112

| access-date = 2024-04-27

}}{{cite journal

| last = Lodge

| first = A.

| title = An approximate expression for the value of 1 + 1/2 + 1/3 + ... + 1/r

| journal = Messenger of Mathematics

| volume = 30

| year = 1904

| pages = 103–107

| url = https://books.google.com/books?id=K4daAAAAYAAJ&dq=%22An%20approximate%20expression%20for%20the%20value%20of%201%2B%22&pg=PA103

}} One of the earliest of these is a formula{{Cite arXiv

| last = Villarino

| first = Mark B.

| title = Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number

| year = 2007

| eprint = 0707.3950

| class = math.CA

| quote = It would also be interesting to develop an expansion for n! into powers of m, a new Stirling expansion, as it were.

}} See formula 1.8 on page 3.{{Cite journal

| last = Mortici

| first = Cristinel

| year = 2010

| title = On the Stirling expansion into negative powers of a triangular number

| journal = Math. Commun.

| volume = 15

| pages = 359–364

| url = https://www.researchgate.net/publication/228562533

| doi =

}} for the {{nowrap|$n$th}} harmonic number attributed to Srinivasa Ramanujan where $\backslash gamma$ is related to $\backslash textstyle\; \backslash ln\; 2T\_\{k\}$ in a series that considers the powers of $\backslash textstyle\; \backslash frac\{1\}\{T\_\{k\}\}$ (an earlier, less-generalizable proof{{Cite journal |last=Cesàro |first=E. |title=Sur la série harmonique |journal=Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale |volume=4 |pages=295–296 |year=1885 |url=http://eudml.org/doc/100057 |language=fr |publisher=Carilian-Goeury et Vor Dalmont}}{{cite book

| last = Bromwich

| first = Thomas John I'Anson

| title = An Introduction to the Theory of Infinite Series

| publisher = American Mathematical Society

| year = 2005

| orig-date = 1908

| edition = 3rd

| location = United Kingdom

| url = https://www.dbraulibrary.org.in/RareBooks/An%20introduction%20to%20the%20theory%20of%20infinite%20series.pdf

| page = 460

}} See exercise 18. by Ernesto Cesàro gives the first two terms of the series, with an error term):

:$\backslash begin\{align\}$

\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{align}

From Stirling's approximation{{cite book

| last1 = Whittaker

| first1 = E.

| last2 = Watson

| first2 = G.

| title = A Course of Modern Analysis

| edition = 5th

| orig-date = 1902

| year = 2021

| page = 271, 275

| isbn = 9781316518939

| doi = 10.1017/9781009004091

}} See Examples 12.21 and 12.50 for exercises on the derivation of the integral form $\backslash textstyle\; \backslash int\_\{-1\}^\{0\}\; \backslash ln\backslash Gamma(z+1)\backslash ,dz$ of the series $\backslash textstyle\; \backslash sum\_\{k=1\}^\{n\}\; \backslash frac\{\backslash zeta(k)\}\{110\_\{k\}\}\; =\; \backslash ln(\backslash sqrt\{2\backslash pi\})$. follows a similar series:

:$\backslash gamma\; =\; \backslash ln\; 2\backslash pi\; -\; \backslash sum\_\{k=2\}^\{\backslash infty\}\; \backslash frac\{\backslash zeta(k)\}\{T\_\{k\}\}$

The series of inverse triangular numbers also features in the study of the Basel problem{{sfn|Lagarias|2013|p=13}}{{cite journal |last=Nelsen |first=R. B. |title=Proof without Words: Sum of Reciprocals of Triangular Numbers |journal=Mathematics Magazine |volume=64 |issue=3 |year=1991 |pages=167|doi=10.1080/0025570X.1991.11977600 }} posed by Pietro Mengoli. Mengoli proved that $\backslash textstyle\; \backslash sum\_\{k\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{2T\_k\}\; =\; 1$, a result Jacob Bernoulli later used to estimate the value of $\backslash zeta(2)$, placing it between $1$ and $\backslash textstyle\; \backslash sum\_\{k\; =\; 1\}^\backslash infty\; \backslash frac\{2\}\{2T\_k\}\; =\; \backslash sum\_\{k\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{T\_\{k\}\}\; =\; 2$. This identity appears in a formula used by Bernhard Riemann to compute roots of the zeta function,{{Cite book | last = Edwards | first = H. M. | title = Riemann's Zeta Function | publisher = Academic Press | year = 1974 | series = Pure and Applied Mathematics, Vol. 58 | pages = 67, 159}} where $\backslash gamma$ is expressed in terms of the sum of roots $\backslash rho$ plus the difference between Boya's expansion and the series of exact unit fractions $\backslash textstyle\; \backslash sum\_\{k\; =\; 1\}^\{\backslash infty\}\; \backslash frac\{1\}\{T\_\{k\}\}$:

:$\backslash gamma\; -\; \backslash ln\; 2\; =\; \backslash ln\; 2\backslash pi\; +\; \backslash sum\_\{\backslash rho\}\; \backslash frac\{2\}\{\backslash rho\}\; -\; \backslash sum\_\{k\; =\; 1\}^\{\backslash infty\}\; \backslash frac\{1\}\{T\_k\}$

{{mvar|γ}} equals the value of a number of definite integrals:

$$\backslash begin\{align\}$$

\gamma &= - \int_0^\infty e^{-x} \log x \,dx \\

&= -\int_0^1\log\left(\log\frac 1 x \right) dx \\

&= \int_0^\infty \left(\frac1{e^x-1}-\frac1{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(\frac1{\log x} + \frac1{1-x}\right)dx\\

&= \int_0^\infty \left(\frac1{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^2x} \, 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 \\

&= \frac{1}{2}+\int_{0}^{\infty}\frac{2x\,dx}{(x^2+1)(e^{2\pi x}-1)}

\end{align}

where {{math|{{var|H}}{{sub|{{var|x}}}}}} is the fractional harmonic number, and $\backslash \{1/x\backslash \}$ is the fractional part of $1/x$.

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

$$\backslash begin\{align\}$$

&\int_0^{\infty} \left(\frac{1}{e^x - 1} - \frac{1}{x e^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{align}

The integral on the second line of the equation is the definition of the Riemann zeta function, which is {{math|{{var|m}}!{{var|ζ}}({{var|m}} + 1)}}.

Definite integrals in which {{mvar|γ}} appears include:{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant |url=https://mathworld.wolfram.com/Euler-MascheroniConstant.html |access-date=2024-10-19 |website=mathworld.wolfram.com |language=en}}{{Cite journal |last=Blagouchine |first=Iaroslav V. |date=2014-10-01 |title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results |url=https://link.springer.com/article/10.1007/s11139-013-9528-5 |journal=The Ramanujan Journal |language=en |volume=35 |issue=1 |pages=21–110 |doi=10.1007/s11139-013-9528-5 |issn=1572-9303}}

$$\backslash begin\{align\}$$

\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 &= \frac12 \log^2 2 - \gamma \end{align}

We also have Catalan's 1875 integral{{r|SondowZudilin2006}}

$$\backslash gamma\; =\; \backslash int\_0^1\; \backslash left(\backslash frac1\{1+x\}\backslash sum\_\{n=1\}^\backslash infty\; x^\{2^n-1\}\backslash right)\backslash ,dx.$$

One can express {{mvar|γ}} using a special case of Hadjicostas's formula as a double integral{{r|Sondow2003a|Sondow2005}} with equivalent series:

$$\backslash begin\{align\}$$

\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{align}

An interesting comparison by Sondow{{r|Sondow2005}} is the double integral and alternating series

$$\backslash begin\{align\}$$

\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{align}

It shows that {{math|log {{sfrac|4|π}}}} may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series{{r|Sondow2005a}}

$$\backslash begin\{align\}$$

\gamma &= \sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} \\

\log\frac4{\pi} &= \sum_{n=1}^\infty \frac{N_1(n) - N_0(n)}{2n(2n+1)} ,

\end{align}

where {{math|{{var|N}}{{sub|1}}({{var|n}})}} and {{math|{{var|N}}{{sub|0}}({{var|n}})}} are the number of 1s and 0s, respectively, in the base 2 expansion of {{mvar|n}}.

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 {{math|{{var|α}} > −{{var|n}}}}. However, the rate of convergence of this expansion depends significantly on {{mvar|α}}. In particular, {{math|{{var|γ}}{{sub|{{var|n}}}}(1/2)}} exhibits much more rapid convergence than the conventional expansion {{math|{{var|γ}}{{sub|{{var|n}}}}(0)}}.{{r|DeTemple1993}}{{sfn|Havil|2003|pp=75–8}} 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 {{mvar|γ}}:

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

The series for {{mvar|γ}} is equivalent to a series Nielsen found in 1897:{{r|Krämer2005}}{{sfn|Blagouchine|2016}}

$$\backslash gamma\; =\; 1\; -\; \backslash sum\_\{k=2\}^\backslash infty\; (-1)^k\backslash frac\{\backslash left\backslash lfloor\backslash log\_2\; k\backslash right\backslash rfloor\}\{k+1\}.$$

In 1910, Vacca found the closely related series{{r|Vacca1910|Glaisher1910|Hardy1912|Vacca1926|Kluyver1927|Krämer2005|Blagouchine2016}}

$$\backslash begin\{align\}$$

\gamma & = \sum_{k=1}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor} k \\[5pt]

& = \tfrac12-\tfrac13 + 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right) + 3\left(\tfrac18 - \tfrac19 + \tfrac1{10} - \tfrac1{11} + \cdots - \tfrac1{15}\right) + \cdots,

\end{align}

where {{math|log{{sub|2}}}} is the logarithm to base 2 and {{math|{{floor| }}}} is the floor function.

This can be generalized to:{{Citation |last1=Pilehrood |first1=Khodabakhsh Hessami |title=Vacca-type series for values of the generalized-Euler-constant function and its derivative |date=2008-08-04 |last2=Pilehrood |first2=Tatiana Hessami|arxiv=0808.0410 }}

$$\backslash gamma=\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{\backslash left\backslash lfloor\backslash log\_B\; k\backslash right\backslash rfloor\}\{k\}\; \backslash varepsilon(k)$$where:

In 1926 Vacca found a second series:

$$\backslash begin\{align\}$$

\gamma + \zeta(2) & = \sum_{k=2}^\infty \left( \frac1{\left\lfloor\sqrt{k}\right\rfloor^2} - \frac1{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]

&= \frac12 + \frac23 + \frac1{2^2}\sum_{k=1}^{2\cdot 2} \frac{k}{k+2^2} + \frac1{3^2}\sum_{k=1}^{3\cdot 2} \frac{k}{k+3^2} + \cdots

\end{align}

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

$$\backslash gamma\; =\; \backslash log\backslash pi\; -\; 4\backslash log\backslash left(\backslash Gamma(\backslash tfrac34)\backslash right)\; +\; \backslash frac\; 4\; \backslash pi\; \backslash sum\_\{k=1\}^\backslash infty\; (-1)^\{k+1\}\backslash frac\{\backslash log(2k+1)\}\{2k+1\}.$$

Ramanujan, in his lost notebook gave a series that approaches {{mvar|γ}}{{r|Berndt2008}}:

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

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

$$\backslash gamma\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac$$

where {{math|{{var|G}}{{sub|{{var|n}}}}}} are Gregory coefficients.{{r|Krämer2005|Blagouchine2016|Blagouchine2018}} This series is the special case {{math|1={{var|k}} = 1}} of the expansions

$$\backslash begin\{align\}$$

\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 + \frac1{2k} + \frac1{12k(k+1)} + \frac1{12k(k+1)(k+2)} + \frac{19}{120k(k+1)(k+2)(k+3)} + \cdots &&

\end{align}

convergent for {{math|1={{var|k}} = 1, 2, ...}}

A similar series with the Cauchy numbers of the second kind {{math|{{var|C}}{{sub|{{var|n}}}}}} is{{r|Blagouchine2016|Alabdulmohsin2018_1478}}

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

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

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

\quad a>-1

where {{math|{{var|ψ}}{{sub|{{var|n}}}}({{var|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 {{mvar|a}} this series contains rational terms only. For example, at {{math|1={{var|a}} = 1}}, it becomes{{r|OEIS_A302120|OEISA302121}}

$$\backslash gamma=\backslash frac\{3\}\{4\}\; -\; \backslash frac\{11\}\{96\}\; -\; \backslash frac\{1\}\{72\}\; -\; \backslash frac\{311\}\{46080\}\; -\; \backslash frac\{5\}\{1152\}\; -\; \backslash frac\{7291\}\{2322432\}\; -\; \backslash frac\{243\}\{100352\}\; -\; \backslash ldots$$

Other series with the same polynomials include these examples:

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

and

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

where {{math|Γ({{var|a}})}} is the gamma function.{{r|Blagouchine2018}}

A series related to the Akiyama–Tanigawa algorithm is

$$\backslash gamma=\; \backslash log(2\backslash pi)\; -\; 2\; -\; 2\; \backslash sum\_\{n=1\}^\backslash infty\backslash frac\{(-1)^n\; G\_\{n\}(2)\}\{n\}=$$

\log(2\pi) - 2 + \frac{2}{3} + \frac{1}{24}+ \frac{7}{540} + \frac{17}{2880}+ \frac{41}{12600} + \ldots

where {{math|{{var|G}}{{sub|{{var|n}}}}(2)}} are the Gregory coefficients of the second order.{{r|Blagouchine2018}}

As a series of prime numbers:

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

{{mvar|γ}} equals the following asymptotic formulas (where {{math|{{var|H}}{{sub|{{var|n}}}}}} is the {{mvar|n}}th harmonic number):

• $\backslash gamma\; \backslash sim\; H\_n\; -\; \backslash log\; n\; -\; \backslash frac1\{2n\}\; +\; \backslash frac1\{12n^2\}\; -\; \backslash frac1\{120n^4\}\; +\; \backslash cdots$ (Euler)
• $\backslash gamma\; \backslash sim\; H\_n\; -\; \backslash log\backslash left(\{n\; +\; \backslash frac1\{2\}\; +\; \backslash frac1\{24n\}\; -\; \backslash frac1\{48n^2\}\; +\; \backslash cdots\}\backslash right)$ (Negoi)
• $\backslash gamma\; \backslash sim\; H\_n\; -\; \backslash frac\{\backslash log\; n\; +\; \backslash log(n+1)\}\{2\}\; -\; \backslash frac1\{6n(n+1)\}\; +\; \backslash frac1\{30n^2(n+1)^2\}\; -\; \backslash cdots$ (Cesàro)

The third formula is also called the Ramanujan expansion.

Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations.{{r|Alabdulmohsin2018_1478}} He showed that (Theorem A.1):

$$\backslash begin\{align\}$$

\sum_{n=1}^\infty \Big(\log n +\gamma - H_n + \frac{1}{2n}\Big) &= \frac{\log (2\pi)-1-\gamma}{2} \\

\sum_{n=1}^\infty \Big(\log \sqrt{n(n+1)} +\gamma - H_n \Big) &= \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{align}

The constant {{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} is important in number theory. Its numerical value is:{{r|OEIS_A073004}}

{{block indent|{{gaps|1.78107|24179|90197|98523|65041|03107|17954|91696|45214|30343|...}}.}}

{{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} equals the following limit, where {{math|{{var|p}}{{sub|{{var|n}}}}}} is the {{mvar|n}}th prime number:

$$e^\backslash gamma\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash frac1\{\backslash log\; p\_n\}\; \backslash prod\_\{i=1\}^n\; \backslash frac\{p\_i\}\{p\_i-1\}.$$

This restates the third of Mertens' theorems.{{r|excursions}}

We further have the following product involving the three constants {{math|{{var|e}}}}, {{math|{{var|π}}}} and {{math|{{var|γ}}}}:{{Cite web |last=Weisstein |first=Eric W. |title=Mertens Theorem |url=https://mathworld.wolfram.com/MertensTheorem.html |access-date=2024-10-08 |website=mathworld.wolfram.com |language=en}}

$$\backslash frac\{\backslash pi^2\}\{6e^\backslash gamma\}=\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash log\; p\_n\; \backslash prod\_\{i=1\}^n\; \backslash frac\{p\_i\}\{p\_i+1\}.$$

Other infinite products relating to {{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} include:

$$\backslash begin\{align\}$$

\frac{e^{1+\frac{\gamma}{2}}}{\sqrt{2\pi}} &= \prod_{n=1}^\infty e^{-1+\frac1{2n}}\left(1+\frac1{n}\right)^n \\

\frac{e^{3+2\gamma}}{2\pi} &= \prod_{n=1}^\infty e^{-2+\frac2{n}}\left(1+\frac2{n}\right)^n. \end{align}

These products result from the Barnes G-function.

In addition,

$$e^\{\backslash gamma\}\; =\; \backslash sqrt\{\backslash frac2\{1\}\}\; \backslash cdot\; \backslash sqrt[3]\{\backslash frac\{2^2\}\{1\backslash cdot\; 3\}\}\; \backslash cdot\; \backslash sqrt[4]\{\backslash frac\{2^3\backslash cdot\; 4\}\{1\backslash cdot\; 3^3\}\}\; \backslash cdot\; \backslash sqrt[5]\{\backslash frac\{2^4\backslash cdot\; 4^4\}\{1\backslash cdot\; 3^6\backslash cdot\; 5\}\}\; \backslash cdots$$

where the {{mvar|n}}th factor is the {{math|({{var|n}} + 1)}}th root of

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

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using hypergeometric functions.{{r|Sondow2003}}

It also holds that{{r|ChoiSrivastava2010}}

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

{{Main|Stieltjes constants}}

File:Generalisation of Euler–Mascheroni constant.jpg

Euler's generalized constants are given by

$$\backslash gamma\_\backslash alpha\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash left(\backslash sum\_\{k=1\}^n\; \backslash frac1\{k^\backslash alpha\}\; -\; \backslash int\_1^n\; \backslash frac1\{x^\backslash alpha\}\backslash ,dx\backslash right)$$

for {{math|0 < {{var|α}} < 1}}, with {{mvar|γ}} as the special case {{math|1={{var|α}} = 1}}.{{sfn|Havil|2003|pp=117–18}} Extending for {{math| {{var|α}} > 1}} gives:

$$\backslash gamma\_\{\backslash alpha\}\; =\; \backslash zeta(\backslash alpha)\; -\; \backslash frac1\{\backslash alpha-1\}$$

with again the limit:

$$\backslash gamma\; =\; \backslash lim\_\{a\backslash to\; 1\}\backslash left(\backslash zeta(a)\; -\; \backslash frac1\{a-1\}\backslash right)$$

This can be further generalized to

$$c\_f\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash left(\backslash sum\_\{k=1\}^n\; f(k)\; -\; \backslash int\_1^n\; f(x)\backslash ,dx\backslash right)$$

for some arbitrary decreasing function {{mvar|f}}. Setting

$$f\_n(x)\; =\; \backslash frac\{(\backslash log\; x)^n\}\{x\}$$

gives rise to the Stieltjes constants $\backslash gamma\_n$, that occur in the Laurent series expansion of the Riemann zeta function:

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

with $\backslash gamma\_0\; =\; \backslash gamma\; =\; 0.577\backslash dots$

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

modulo class:{{r|RamMurtySaradha2010}}

The basic properties are

$$\backslash begin\{align\}$$

&\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{align}

and if the greatest common divisor {{math|1=gcd({{var|a}},{{var|q}}) = {{var|d}}}} then

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

A two-dimensional generalization of Euler's constant is the Masser-Gramain constant. It is defined as the following limiting difference:{{Cite web |last=Weisstein |first=Eric W. |title=Masser-Gramain Constant |url=https://mathworld.wolfram.com/Masser-GramainConstant.html |access-date=2024-10-19 |website=mathworld.wolfram.com |language=en}}

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

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: .{{Cite web |last1=Melquiond |first1=Guillaume |last2=Nowak |first2=W. Georg |last3=Zimmermann |first3=Paul |title=Numerical approximation of the Masser-Gramain constant to four decimal digits |url=https://www.lri.fr/~melquion/doc/12-mc.pdf |access-date=2024-10-03}}

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

• {{cite journal |last=Bretschneider |first=Carl Anton |date=1837 |orig-date=1835 |title=Theoriae logarithmi integralis lineamenta nova |journal=Crelle's Journal |volume=17 |pages=257–285 |language=la |url=https://zenodo.org/record/1448830

}}

• {{cite book |last=Havil |first=Julian |date=2003 |title=Gamma: Exploring Euler's Constant |publisher=Princeton University Press |isbn=978-0-691-09983-5

}}

• {{cite journal |last=Lagarias |first=Jeffrey C. |date=2013 |title=Euler's constant: Euler's work and modern developments |journal=Bulletin of the American Mathematical Society |volume=50 |issue=4 |page=556 |doi=10.1090/s0273-0979-2013-01423-x |arxiv=1303.1856 |s2cid=119612431

}}

{{Reflist|refs=

{{Cite OEIS |A002852 |Continued fraction for Euler's constant}}

{{cite book |last=De Morgan |first=Augustus |author-link=Augustus De Morgan |date=1836–1842 |title=The differential and integral calculus |location=London |publisher=Baldwin and Craddoc |at="{{math|γ}}" on [https://books.google.com/books?id=95x4IrIcHrgC&pg=PA578 p. 578] }}

{{cite book |last1=Caves |first1=Carlton M. |author-link1=Carlton M. Caves |last2=Fuchs |first2=Christopher A. |date=1996 |chapter=Quantum information: How much information in a state vector? |title=The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later |publisher=Israel Physical Society |isbn=9780750303941 |oclc=36922834 |arxiv=quant-ph/9601025 |bibcode=1996quant.ph..1025C}}

{{Cite book |last1=Haible |first1=Bruno |last2=Papanikolaou |first2=Thomas |title=Algorithmic Number Theory |chapter=Fast multiprecision evaluation of series of rational numbers |date=1998 |editor-last=Buhler |editor-first=Joe P. |series=Lecture Notes in Computer Science |volume=1423 |publisher=Springer |pages=338–350 |doi=10.1007/bfb0054873 |isbn=9783540691136}}

{{cite thesis |last=Papanikolaou |first=T. |date=1997 |title=Entwurf und Entwicklung einer objektorientierten Bibliothek für algorithmische Zahlentheorie |publisher=Universität des Saarlandes |url=https://www-old.cdc.informatik.tu-darmstadt.de/reports/reports/papa.diss.ps.gz |language=de}}

See also {{cite journal |last1=Sondow |first1=Jonathan |date=2003 |title=Criteria for irrationality of Euler's constant |journal=Proceedings of the American Mathematical Society |volume=131 |issue=11 |pages=3335–3344 |doi=10.1090/S0002-9939-03-07081-3 |s2cid=91176597 |arxiv=math.NT/0209070 }}

{{Cite journal |last1=Mahler |first1=Kurt |date=4 June 1968 |title=Applications of a theorem by A. B. Shidlovski |journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |volume=305 |issue=1481 |pages=149–173 |doi=10.1098/rspa.1968.0111 |bibcode=1968RSPSA.305..149M |s2cid=123486171|url=https://carmamaths.org/resources/mahler/docs/169.pdf}}

{{Cite journal |last1=Ram Murty |first1=M. |first2=N. |last2=Saradha |date=2010 |title=Euler–Lehmer constants and a conjecture of Erdos |journal=Journal of Number Theory |volume=130 |issue=12 |pages=2671–2681 |doi=10.1016/j.jnt.2010.07.004 |issn=0022-314X |doi-access=free }}

{{cite arXiv |last=Aptekarev |first=A. I. |date=28 February 2009 |title=On linear forms containing the Euler constant |class=math.NT |eprint=0902.1768}}

{{Cite journal |last=Rivoal |first=Tanguy |date=2012 |title=On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant |journal=Michigan Mathematical Journal |volume=61 |issue=2 |pages=239–254 |doi=10.1307/mmj/1339011525 |issn=0026-2285 |doi-access=free |url=https://projecteuclid.org/euclid.mmj/1339011525}}

{{Cite journal |last1=Murty |first1=M. Ram |last2=Zaytseva |first2=Anastasia |date=2013 |title=Transcendence of Generalized Euler Constants |journal=The American Mathematical Monthly |volume=120 |issue=1 |pages=48–54 |doi=10.4169/amer.math.monthly.120.01.048 |jstor=10.4169/amer.math.monthly.120.01.048 |s2cid=20495981 |issn=0002-9890 |url=https://www.jstor.org/stable/10.4169/amer.math.monthly.120.01.048 }}

{{cite book |last1=Krämer |first1= Stefan |date=2005 |title=Die Eulersche Konstante {{math|γ}} und verwandte Zahlen |publisher=University of Göttingen |language=de }}

{{cite journal |last1=Sondow |first1=Jonathan |date=1998 |title=An antisymmetric formula for Euler's constant |journal=Mathematics Magazine |volume=71 |issue=3 |pages=219–220 |doi=10.1080/0025570X.1998.11996638 |access-date=2006-05-29 |url=http://home.earthlink.net/~jsondow/id8.html |archive-date=2011-06-04 |archive-url=https://web.archive.org/web/20110604123534/http://home.earthlink.net/~jsondow/id8.html}}

{{cite journal |last=Sondow |first=Jonathan |date=2005 |title=Double integrals for Euler's constant and $\backslash log\; \backslash frac\{4\}\{\backslash pi\}$ and an analog of Hadjicostas's formula |journal=American Mathematical Monthly |volume=112 |issue=1 |pages=61–65 |doi=10.2307/30037385 |jstor=30037385 |arxiv=math.CA/0211148}}

{{cite book |last1= Sondow |first1=Jonathan |date=1 August 2005a |title=New Vacca-type rational series for Euler's constant and its 'alternating' analog $\backslash log\; \backslash frac\{4\}\{\backslash pi\}$ |arxiv=math.NT/0508042}}

{{cite journal |last1=Sondow |first1=Jonathan |first2=Wadim |last2=Zudilin |date=2006 |title=Euler's constant, {{math|q}}-logarithms, and formulas of Ramanujan and Gosper |journal=The Ramanujan Journal |volume=12 |issue=2 |pages=225–244 |doi=10.1007/s11139-006-0075-1 |s2cid=1368088 |arxiv=math.NT/0304021}}

{{cite journal |last=DeTemple |first=Duane W. |date=May 1993 |title=A Quicker Convergence to Euler's Constant |journal=The American Mathematical Monthly |volume=100 |issue=5 |pages=468–470 |doi=10.2307/2324300 |issn=0002-9890 |jstor=2324300}}

{{cite journal |last=Blagouchine |first=Iaroslav V. |date=2016 |title=Expansions of generalized Euler's constants into the series of polynomials in {{math|π⁻²}} and into the formal enveloping series with rational coefficients only |journal=J. Number Theory |volume=158 |pages=365–396 |doi=10.1016/j.jnt.2015.06.012 |arxiv=1501.00740}}

{{cite journal |last1=Vacca |first1=G. |author-link=Giovanni Vacca (mathematician) |date=1910 |title=A new analytical expression for the number π and some historical considerations |journal=Bulletin of the American Mathematical Society |volume=16 |pages=368–369|doi=10.1090/S0002-9904-1910-01919-4 |doi-access=free }}

{{cite journal |last1=Vacca |first1=G. |author-link=Giovanni Vacca (mathematician) |date=1926 |title=Nuova serie per la costante di Eulero, {{math|C}} = 0,577...". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche |journal=Matematiche e Naturali |volume=6 |issue=3 |pages=19–20 |language=it}}

{{cite journal |last=Glaisher |first=James Whitbread Lee |author-link=James Whitbread Lee Glaisher |date=1910 |title=On Dr. Vacca's series for {{math|γ}} |journal=Q. J. Pure Appl. Math. |volume=41 |pages=365–368}}

{{ cite journal |last1= Hardy |first1=G.H. |date=1912 |title=Note on Dr. Vacca's series for {{math|γ}} |journal=Q. J. Pure Appl. Math. |volume=43 |pages=215–216}}

{{cite journal |last1=Kluyver |first1=J.C. |date=1927 |title=On certain series of Mr. Hardy |journal=Q. J. Pure Appl. Math. |volume=50 |pages=185–192}}

{{cite journal |last1=Blagouchine |first1=Iaroslav V. |date=2018 |title=Three notes on Ser's and Hasse's representations for the zeta-functions |journal=INTEGERS: The Electronic Journal of Combinatorial Number Theory |volume=18A |issue=#A3 |pages=1–45 |doi=10.5281/zenodo.10581385 |bibcode=2016arXiv160602044B |arxiv=1606.02044 |url=http://math.colgate.edu/~integers/vol18a.html}}

{{cite book |last=Alabdulmohsin |first=Ibrahim M. |date=2018 |title=Summability Calculus. A Comprehensive Theory of Fractional Finite Sums |publisher=Springer |pages=147–8 |isbn=9783319746487}}

{{cite OEIS|A302120|Absolute value of the numerators of a series converging to Euler's constant}}

{{cite OEIS|A302121|Denominators of a series converging to Euler's constant}}

{{cite book

| last = Ramaré | first = Olivier

| doi = 10.1007/978-3-030-73169-4

| isbn = 978-3-030-73168-7

| location = Basel

| mr = 4400952

| page = 131

| publisher = Birkhäuser/Springer

| series = Birkhäuser Advanced Texts: Basel Textbooks

| title = Excursions in Multiplicative Number Theory

| url = https://books.google.com/books?id=n1piEAAAQBAJ&pg=PA131

| year = 2022| s2cid = 247271545

}}

{{cite OEIS|A073004|Decimal expansion of exp(gamma)}}

{{Cite journal |last1=Choi |first1=Junesang |last2=Srivastava |first2=H.M. |date=1 September 2010 |title=Integral Representations for the Euler–Mascheroni Constant {{math|γ}} |journal=Integral Transforms and Special Functions |volume=21 |issue=9 |pages=675–690 |doi=10.1080/10652461003593294 |s2cid=123698377 |issn=1065-2469}}

{{cite arXiv |last1=Sondow |first1=Jonathan |date=2003 |eprint=math.CA/0306008 |title= An infinite product for {{math|e{{sup|γ}}}} via hypergeometric formulas for Euler's constant, {{math|γ}}}}

{{cite journal |last1=Knuth |first1=Donald E. |author-link=Donald Knuth |date=July 1962 |title=Euler's Constant to 1271 Places |journal=Mathematics of Computation |volume=16 |issue=79 |pages=275–281 |publisher=American Mathematical Society |doi=10.2307/2004048 |jstor=2004048 |url=https://www.jstor.org/stable/2004048}}

{{cite web |last=Yee |first=Alexander J. |date=7 March 2011 |title=Large Computations |website=www.numberworld.org |url=http://www.numberworld.org/nagisa_runs/computations.html}}

{{cite web |last=Yee |first=Alexander J. |title=Records Set by y-cruncher |website=www.numberworld.org |access-date=30 April 2018 |url=http://www.numberworld.org/y-cruncher/records.html }}

{{cite web |last=Yee |first=Alexander J. |title=y-cruncher - A Multi-Threaded Pi-Program |website=www.numberworld.org |url=http://www.numberworld.org/y-cruncher/}}

{{cite web |title=Euler-Mascheroni Constant |website=Polymath Collector |date=15 February 2020 |url=https://ehfd.github.io/world-record/euler-mascheroni-constant/}}

{{cite journal

| last1 = Connallon | first1 = Tim

| last2 = Hodgins | first2 = Kathryn A.

| date = October 2021

| doi = 10.1111/evo.14372

| issue = 11

| journal = Evolution

| pages = 2624–2640

| title = Allen Orr and the genetics of adaptation

| volume = 75| pmid = 34606622

| s2cid = 238357410

}}

{{cite journal |last1=Berndt |first1=Bruce C.|author-link=Bruce C. Berndt |date=January 2008 |title=A fragment on Euler's constant in Ramanujan's lost notebook |journal=South East Asian J. Math. & Math. Sc |volume=6 |issue=2 |pages=17–22 |doi= |jstor= |url=https://scholarworks.utrgv.edu/cgi/viewcontent.cgi?article=1130&context=mss_fac}}

{{Cite OEIS|A001620|Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma}}

}}

• {{cite journal |author1=Borwein, Jonathan M. |author2=David M. Bradley |author3=Richard E. Crandall |title=Computational Strategies for the Riemann Zeta Function |journal=Journal of Computational and Applied Mathematics |date=2000 |volume=121 |issue=1–2 |pages=11 |doi=10.1016/s0377-0427(00)00336-8 |bibcode=2000JCoAM.121..247B |ref=none |doi-access=free }} Derives {{math|γ}} as sums over Riemann zeta functions.
• {{cite book |last=Finch |first=Steven R. |date=2003 |title=Mathematical Constants|series= Encyclopedia of Mathematics and its Applications|volume=94|publisher=Cambridge University Press|location=Cambridge|isbn=0-521-81805-2|ref=none}}
• {{cite journal |first1=I. |last1=Gerst |title=Some series for Euler's constant |date=1969 |journal=Amer. Math. Monthly |doi=10.2307/2316370 |volume=76 |issue=3 |pages=237–275 |jstor=2316370 |ref=none}}
• {{cite journal |last=Glaisher |first=James Whitbread Lee |author-link=James Whitbread Lee Glaisher |date=1872 |title=On the history of Euler's constant |journal=Messenger of Mathematics |volume=1 |pages=25–30 |jfm=03.0130.01 |ref=none}}
• {{cite web |last1=Gourdon|first1=Xavier|last2=Sebah|first2=P. |date=2002 |title=Collection of formulae for Euler's constant, {{math|γ}} |url=http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html |ref=none}}
• {{cite web |last1=Gourdon|first1=Xavier|last2=Sebah|first2=P. |date=2004 |title=The Euler constant: {{math|γ}} |url=http://numbers.computation.free.fr/Constants/Gamma/gamma.html |ref=none}}
• Julian Havil (2003): GAMMA: Exploring Euler's Constant, Princeton University Press, ISBN 978-0-69114133-6.
• {{cite journal |first1=E. A. |last1= Karatsuba |title=Fast evaluation of transcendental functions |journal=Probl. Inf. Transm. |volume =27 |number=44 |pages=339–360 |date=1991 |ref=none}}
• {{cite journal |last1=Karatsuba |first1=E.A. |date=2000 |title=On the computation of the Euler constant {{math|γ}} |journal=Journal of Numerical Algorithms |volume=24 |issue=1–2 |pages=83–97 |doi=10.1023/A:1019137125281 |s2cid=21545868 |ref=none}}
• {{cite book |last=Knuth |first=Donald |author-link=Donald Knuth |date=1997 |title=The Art of Computer Programming, Vol. 1 |edition=3rd |publisher=Addison-Wesley |isbn=0-201-89683-4 |pages=75, 107, 114, 619–620 |ref=none}}
• {{cite journal |first1=D. H. |last1=Lehmer |date=1975 |title=Euler constants for arithmetical progressions |journal=Acta Arith. |volume=27 |number=1 |pages=125–142 |url=http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf |doi=10.4064/aa-27-1-125-142 |ref=none|doi-access=free }}
• {{cite journal |last1=Lerch |first1=M. |date=1897 |title=Expressions nouvelles de la constante d'Euler |journal=Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften |volume=42 |page=5 |ref=none}}
• {{cite book |last=Mascheroni |first=Lorenzo |author-link=Lorenzo Mascheroni |date=1790 |title=Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur |publisher=Galeati, Ticini |ref=none}}
• {{cite journal |last1=Sondow |first1=Jonathan |date=2002 |title=A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant |arxiv=math.NT/0211075 |journal=Mathematica Slovaca |volume=59 |pages=307–314 |doi=10.2478/s12175-009-0127-2 |bibcode=2002math.....11075S |s2cid=16340929 |ref=none}} with an Appendix by [https://web.archive.org/web/20130523085959/http://wain.mi.ras.ru/zlobin/ Sergey Zlobin]

• {{springer|title=Euler constant|id=p/e036420|mode=cs1}}
• {{MathWorld|urlname=Euler-MascheroniConstant|title=Euler–Mascheroni constant|ref=none}}
• [https://jonathansondow.github.io/ Jonathan Sondow.]
• [http://www.ccas.ru/personal/karatsuba/algen.htm Fast Algorithms and the FEE Method], E.A. Karatsuba (2005)
• Further formulae which make use of the constant: [http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html Gourdon and Sebah (2004).]

{{Leonhard Euler}}

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{{DEFAULTSORT:Euler's constant}}

Category:Mathematical constants

Category:Unsolved problems in number theory

Category:Leonhard Euler