Editing Euler's constant (section)

== Formulas and identities ==

=== Relation to gamma function ===
{{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:

<math display="block">-\gamma = \Gamma'(1) = \Psi(1). </math>

This is equal to the limits:

<math display="block">\begin{align}-\gamma &= \lim_{z\to 0}\left(\Gamma(z) - \frac1{z}\right) \\&= \lim_{z\to 0}\left(\Psi(z) + \frac1{z}\right).\end{align}</math>

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

<math display="block">\begin{align} \lim_{z\to 0}\frac1{z}\left(\frac1{\Gamma(1+z)} - \frac1{\Gamma(1-z)}\right) &= 2\gamma \\
\lim_{z\to 0}\frac1{z}\left(\frac1{\Psi(1-z)} - \frac1{\Psi(1+z)}\right) &= \frac{\pi^2}{3\gamma^2}. \end{align}</math>

A limit related to the [[beta function]] (expressed in terms of [[gamma function]]s) is

<math display="block">\begin{align} \gamma &= \lim_{n\to\infty}\left(\frac{ \Gamma\left(\frac1{n}\right) \Gamma(n+1)\, n^{1+\frac1{n}}}{\Gamma\left(2+n+\frac1{n}\right)} - \frac{n^2}{n+1}\right) \\
&= \lim\limits_{m\to\infty}\sum_{k=1}^m{m \choose k}\frac{(-1)^k}{k}\log\big(\Gamma(k+1)\big). \end{align}</math>

=== Relation to the zeta function ===
{{mvar|γ}} can also be expressed as an [[series (mathematics)|infinite sum]] whose terms involve the [[Riemann zeta function]] evaluated at positive integers:

<math display="block">\begin{align}\gamma &= \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{m} \\
 &= \log\frac4{\pi} + \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{2^{m-1}m}.\end{align} </math>
The constant <math>\gamma</math> can also be expressed in terms of the sum of the reciprocals of [[Riemann hypothesis|non-trivial zeros]] <math>\rho</math> of the zeta function:<ref name="Marek6infinity2019">{{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 <math>\rho</math> and complex conjugate <math>\bar{\rho}</math> are paired together and summed according to increasing absolute values of the imaginary parts of {{nowrap|<math>\rho</math>.}}"}} 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.</ref>
:<math>\gamma = \log 4\pi + \sum_{\rho} \frac{2}{\rho} - 2</math>
Other series related to the zeta function include:

<math display="block">\begin{align} \gamma &= \tfrac3{2}- \log 2 - \sum_{m=2}^\infty (-1)^m\,\frac{m-1}{m}\big(\zeta(m)-1\big) \\
 &= \lim_{n\to\infty}\left(\frac{2n-1}{2n} - \log n + \sum_{k=2}^n \left(\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}</math>

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}}

<math display="block">\begin{align} \gamma &= \lim_{s\to 1^+}\sum_{n=1}^\infty \left(\frac1{n^s}-\frac1{s^n}\right) \\&= \lim_{s\to 1}\left(\zeta(s) - \frac{1}{s-1}\right) \\&= \lim_{s\to 0}\frac{\zeta(1+s)+\zeta(1-s)}{2} \end{align}</math>

and the following formula, established in 1898 by [[Charles Jean de la Vallée-Poussin|de la Vallée-Poussin]]:

<math display="block">\gamma = \lim_{n\to\infty}\frac1{n}\, \sum_{k=1}^n \left(\left\lceil \frac{n}{k} \right\rceil - \frac{n}{k}\right)</math>

where {{math|{{ceil|&nbsp;}}}} are [[ceiling function|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:

<math display="block">\gamma =\lim_{n\to\infty}\left( \sum_{k=1}^n \frac1{k} - \log n -\sum_{m=2}^\infty \frac{\zeta(m,n+1)}{m}\right),</math>

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

<math display="block">H_n = \log(n) + \gamma + \frac1{2n} - \frac1{12n^2} + \frac1{120n^4} - \varepsilon,</math>
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]]:

<math display="block">\gamma =12\,\log(A)-\log(2\pi)+\frac{6}{\pi^2}\,\zeta'(2)</math>

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

<math display="block">\gamma=\lim_{n\to\infty}\left(-n+\zeta\left(\frac{n+1}{n}\right)\right)</math>
=== Relation to triangular numbers ===
Numerous formulations have been derived that express <math>\gamma</math> in terms of sums and logarithms of [[triangular numbers]].<ref name="Boya2008AnotherRelation">{{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.</ref><ref name="Jonathan2005DoubleIntegrals">{{Cite journal
| last = Sondow
| first = Jonathan
| title = Double Integrals for Euler's Constant and <math>\textstyle \frac{4}{\pi}</math> 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
}}</ref><ref>{{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
}}</ref><ref>{{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
}}</ref> One of the earliest of these is a formula<ref>{{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.</ref><ref>{{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 =
}}</ref> for the {{nowrap|<math>n</math>th}} [[harmonic number]] attributed to [[Srinivasa Ramanujan]] where <math>\gamma</math> is related to <math>\textstyle \ln 2T_{k}</math> in a series that considers the powers of <math>\textstyle \frac{1}{T_{k}}</math> (an earlier, less-generalizable proof<ref>{{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}}</ref><ref>{{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.</ref> by [[Ernesto Cesàro]] gives the first two terms of the series, with an error term):

:<math>\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}</math>

From [[Stirling's approximation]]<ref name="Boya2008AnotherRelation"/><ref>{{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 <math>\textstyle \int_{-1}^{0} \ln\Gamma(z+1)\,dz</math> of the series <math>\textstyle \sum_{k=1}^{n} \frac{\zeta(k)}{110_{k}} = \ln(\sqrt{2\pi})</math>.</ref> follows a similar series:

:<math>\gamma = \ln 2\pi - \sum_{k=2}^{\infty} \frac{\zeta(k)}{T_{k}}</math>

The series of inverse triangular numbers also features in the study of the [[Basel problem]]{{sfn|Lagarias|2013|p=13}}<ref>{{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 }}</ref> posed by [[Pietro Mengoli]]. Mengoli proved that <math>\textstyle \sum_{k = 1}^\infty \frac{1}{2T_k} = 1</math>, a result [[Jacob Bernoulli]] later used to estimate the [[Basel_problem#The_Riemann_zeta_function|value]] of <math>\zeta(2)</math>, placing it between <math>1</math> and <math>\textstyle \sum_{k = 1}^\infty \frac{2}{2T_k} = \sum_{k = 1}^\infty \frac{1}{T_{k}} = 2</math>. This identity appears in a formula used by [[Bernhard Riemann]] to compute [[Eulers_constant#Relation_to_the_zeta_function|roots of the zeta function]],<ref>{{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}}</ref> where <math>\gamma</math> is expressed in terms of the sum of roots <math>\rho</math> plus the difference between Boya's expansion and the series of exact [[Unit fraction|unit fractions]] <math>\textstyle \sum_{k = 1}^{\infty} \frac{1}{T_{k}}</math>:

:<math>\gamma - \ln 2 = \ln 2\pi + \sum_{\rho} \frac{2}{\rho} - \sum_{k = 1}^{\infty} \frac{1}{T_k}</math>

=== Integrals ===
{{mvar|γ}} equals the value of a number of definite [[integral]]s:

<math display="block">\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 
 \end{align}
 </math>
where {{math|{{var|H}}{{sub|{{var|x}}}}}} is the [[Harmonic number#Harmonic numbers for real and complex values|fractional harmonic number]], and <math>\{1/x\}</math> is the [[fractional part]] of <math>1/x</math>.

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

<math display="block">\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}</math>

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:<ref name=":3">{{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}}</ref><ref name=":1">{{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}}</ref>

<math display="block">\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}</math>

We also have [[Eugène Charles Catalan|Catalan]]'s 1875 integral{{r|SondowZudilin2006}}

<math display="block">\gamma = \int_0^1 \left(\frac1{1+x}\sum_{n=1}^\infty x^{2^n-1}\right)\,dx.</math>

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

<math display="block">\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}</math>

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

<math display="block">\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}</math>

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}}

<math display="block">\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}</math>

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 [[Binary number|base 2]] expansion of {{mvar|n}}.

=== Series expansions ===
In general,

<math display="block">
\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)
</math>

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

<math display="block">
\frac{1}{2(n+1)} < \gamma_n(0) - \gamma < \frac{1}{2n},
</math>

while

<math display="block">
\frac{1}{24(n+1)^2} < \gamma_n(1/2) - \gamma < \frac{1}{24n^2}.
</math>

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}}:
<math display="block">\gamma = \sum_{k=1}^\infty \left(\frac 1 k - \log\left(1+\frac 1 k \right)\right).</math>

The series for {{mvar|γ}} is equivalent to a series [[Niels Nielsen (mathematician)|Nielsen]] found in 1897:{{r|Krämer2005}}{{sfn|Blagouchine|2016}}

<math display="block">\gamma = 1 - \sum_{k=2}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor}{k+1}.</math>

In 1910, [[Giovanni Vacca (mathematician)|Vacca]] found the closely related series{{r|Vacca1910|Glaisher1910|Hardy1912|Vacca1926|Kluyver1927|Krämer2005|Blagouchine2016}}

<math display="block">\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}</math>

where {{math|log{{sub|2}}}} is the [[binary logarithm|logarithm to base 2]] and {{math|{{floor|&nbsp;}}}} is the [[Floor and ceiling functions|floor function]].

This can be generalized to:<ref>{{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 }}</ref>

<math display="block">\gamma= \sum_{k=1}^\infty \frac{\left\lfloor\log_B k\right\rfloor}{k} \varepsilon(k)</math>where:<math display="block">\varepsilon(k)= \begin{cases} B-1, &\text{if } B\mid n \\ -1, &\text{if }B\nmid n \end{cases}</math>

In 1926 Vacca found a second series:

<math display="block">\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}</math>

From the [[Carl Johan Malmsten|Malmsten]]–[[Ernst Kummer|Kummer]] expansion for the logarithm of the gamma function<ref name=":1" /> we get:

<math display="block">\gamma = \log\pi - 4\log\left(\Gamma(\tfrac34)\right) + \frac 4 \pi \sum_{k=1}^\infty (-1)^{k+1}\frac{\log(2k+1)}{2k+1}.</math>

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

<math display="block">\gamma = \log 2 - \sum_{n=1}^{\infty} \sum_{k=\frac{3^{n-1}+1}{2}}^{\frac{3^{n}-1}{2}} \frac{2n}{(3k)^3-3k}</math>

An important expansion for Euler's constant is due to [[Gregorio Fontana|Fontana]] and [[Lorenzo Mascheroni|Mascheroni]]

<math display="block">\gamma = \sum_{n=1}^\infty \frac{|G_n|}{n} = \frac12 + \frac1{24} + \frac1{72} + \frac{19}{2880} + \frac3{800} + \cdots,</math>
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

<math display="block">\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}</math>

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}} 

<math display="block">\gamma = 1 -  \sum_{n=1}^\infty \frac{C_{n}}{n \, (n+1)!} =1- \frac{1}{4} -\frac{5}{72} - \frac{1}{32} - \frac{251}{14400} - \frac{19}{1728} - \ldots</math>

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

<math display="block">\gamma=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n}\Big\{\psi_{n}(a)+ \psi_{n}\Big(-\frac{a}{1+a}\Big)\Big\},
\quad a>-1</math>

where {{math|{{var|ψ}}{{sub|{{var|n}}}}({{var|a}})}} are the [[Bernoulli polynomials of the second kind]], which are defined by the generating function

<math display="block">
\frac{z(1+z)^s}{\log(1+z)}= \sum_{n=0}^\infty z^n \psi_n(s) ,\qquad |z|<1.
</math>

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}}

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

<math display="block">\gamma= -\log(a+1) - \sum_{n=1}^\infty\frac{(-1)^n \psi_{n}(a)}{n},\qquad \Re(a)>-1 </math>

and

<math display="block">\gamma= -\frac{2}{1+2a} \left\{\log\Gamma(a+1) -\frac{1}{2}\log(2\pi) + \frac{1}{2} + \sum_{n=1}^\infty\frac{(-1)^n \psi_{n+1}(a)}{n}\right\},\qquad \Re(a)>-1 </math>

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

A series related to the Akiyama–Tanigawa algorithm is

<math display="block">\gamma= \log(2\pi) - 2 - 2 \sum_{n=1}^\infty\frac{(-1)^n G_{n}(2)}{n}=
\log(2\pi) - 2 + \frac{2}{3} + \frac{1}{24}+ \frac{7}{540} + \frac{17}{2880}+ \frac{41}{12600} + \ldots  </math>

where {{math|{{var|G}}{{sub|{{var|n}}}}(2)}} are the [[Gregory coefficients]] of the second order.{{r|Blagouchine2018}}
  
As a series of [[prime number]]s:

<math display="block">\gamma = \lim_{n\to\infty}\left(\log n - \sum_{p\le n}\frac{\log p}{p-1}\right).</math>

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

*<math display="inline">\gamma \sim H_n - \log n - \frac1{2n} + \frac1{12n^2} - \frac1{120n^4} + \cdots</math> (''Euler'')
*<math display="inline">\gamma \sim H_n - \log\left({n + \frac1{2} + \frac1{24n} - \frac1{48n^2} + \cdots}\right)</math> (''Negoi'')
*<math display="inline">\gamma \sim H_n - \frac{\log n + \log(n+1)}{2} - \frac1{6n(n+1)} + \frac1{30n^2(n+1)^2} - \cdots</math> (''[[Ernesto Cesàro|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):

<math display="block">\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}</math>

=== Exponential ===
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 of a sequence|limit]], where {{math|{{var|p}}{{sub|{{var|n}}}}}} is the {{mvar|n}}th [[prime number]]:

<math display="block">e^\gamma = \lim_{n\to\infty}\frac1{\log p_n} \prod_{i=1}^n \frac{p_i}{p_i-1}.</math>

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}}}}:<ref name=":9">{{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}}</ref>

<math display="block">\frac{\pi^2}{6e^\gamma}=\lim_{n\to\infty}\frac1{\log p_n} \prod_{i=1}^n \frac{p_i}{p_i+1}.</math>

Other [[infinite product]]s relating to {{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} include:

<math display="block">\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}</math>

These products result from the [[Barnes G-function|Barnes {{mvar|G}}-function]].

In addition,

<math display="block">e^{\gamma} = \sqrt{\frac2{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</math>

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

<math display="block">\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}.</math>

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

It also holds that{{r|ChoiSrivastava2010}}

<math display="block">\frac{e^\frac{\pi}{2}+e^{-\frac{\pi}{2}}}{\pi e^\gamma}=\prod_{n=1}^\infty\left(e^{-\frac{1}{n}}\left(1+\frac{1}{n}+\frac{1}{2n^2}\right)\right).</math>