Editing Euler's constant (section)

==Generalizations==
=== Stieltjes constants ===
{{Main|Stieltjes constants}}
[[File:Generalisation of Euler–Mascheroni constant.jpg|thumb|upright=2|Euler's generalized constants {{math|abm({{var|-<math>\alpha</math>}})}} for {{math|{{var|α}} > 0}}.|alt=]]
''Euler's generalized constants'' are given by

<math display="block">\gamma_\alpha = \lim_{n\to\infty}\left(\sum_{k=1}^n \frac1{k^\alpha} - \int_1^n \frac1{x^\alpha}\,dx\right)</math>

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:

<math display="block">\gamma_{\alpha} = \zeta(\alpha) - \frac1{\alpha-1}</math>

with again the limit:

<math display="block">\gamma = \lim_{a\to 1}\left(\zeta(a) - \frac1{a-1}\right)</math>

This can be further generalized to

<math display="block">c_f = \lim_{n\to\infty}\left(\sum_{k=1}^n f(k) - \int_1^n f(x)\,dx\right)</math>

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

<math display="block">f_n(x) = \frac{(\log x)^n}{x}</math>

gives rise to the [[Stieltjes constants]] <math>\gamma_n</math>, that occur in the [[Laurent series]] expansion of the [[Riemann zeta function]]:

: <math>\zeta(1+s)=\frac{1}{s}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n s^n.</math>

with <math>\gamma_0 = \gamma = 0.577\dots</math>
{| class="wikitable"
|''n''
|approximate value of γ<sub>''n''</sub>
|[[OEIS]]
|-
|0
| +0.5772156649015
|{{OEIS link|A001620}}
|-
|1
|−0.0728158454836
|{{OEIS link|A082633}}
|-
|2
|−0.0096903631928
|{{OEIS link|A086279}}
|-
|3
| +0.0020538344203
|{{OEIS link|A086280}}
|-
|4
| +0.0023253700654
|{{OEIS link|A086281}}
|-
|100
|−4.2534015717080 × 10<sup>17</sup>
|
|-
|1000
|−1.5709538442047 × 10<sup>486</sup>
|
|}

=== Euler-Lehmer constants ===

''Euler–Lehmer constants'' are given by summation of inverses of numbers in a common
modulo class:{{r|RamMurtySaradha2010}}

<math display="block">\gamma(a,q) = \lim_{x\to \infty}\left (\sum_{0<n\le x \atop n\equiv a \pmod q} \frac1{n}-\frac{\log x}{q}\right).</math>

The basic properties are

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

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

<math display="block">q\gamma(a,q) = \frac{q}{d}\gamma\left(\frac{a}{d},\frac{q}{d}\right)-\log d.</math>

===Masser-Gramain constant===

A two-dimensional generalization of Euler's constant is the [[Masser-Gramain constant]]. It is defined as the following limiting difference:<ref>{{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}}</ref>

:<math>\delta = \lim_{n\to\infty} \left( -\log n + \sum_{k=2}^n \frac{1}{\pi r_k^2} \right)</math>

where <math>r_k</math> is the smallest radius of a disk in the complex plane containing at least <math>k</math> [[Gaussian integer|Gaussian integers]].

The following bounds have been established: <math>1.819776 < \delta < 1.819833</math>.<ref>{{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}}</ref>