Editing Euler's constant (section)

=== 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>