Английская Википедия:Euler's constant

The area of the blue region converges to Euler's constant.

Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (Шаблон:Math), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by Шаблон:Math:

<math display="block">\begin{align} \gamma &= \lim_{n\to\infty}\left(-\log n + \sum_{k=1}^n \frac1{k}\right)\\[5px] &=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx. \end{align}</math>

Here, Шаблон:Math represents the floor function.

The numerical value of Euler's constant, to 50 decimal places, is:Шаблон:R

Шаблон:Block indent

Шаблон:Unsolved

History

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations Шаблон:Math and Шаблон:Math for the constant. In 1790, the Italian mathematician Lorenzo Mascheroni used the notations Шаблон:Math and Шаблон:Math for the constant. The notation Шаблон:Math 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.Шаблон:R For example, the German mathematician Carl Anton Bretschneider used the notation Шаблон:Math in 1835,Шаблон:Sfn and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.Шаблон:R

Appearances

Euler's constant appears, among other places, in the following (where '*' means that this entry contains an explicit equation):

Expressions involving the exponential integral*
The Laplace transform* of the natural logarithm
The first term of the Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants*
Calculations of the digamma function
A product formula for the gamma function
The asymptotic expansion of the gamma function for small arguments.
An inequality for Euler's totient function
The growth rate of the divisor function
In dimensional regularization of Feynman diagrams in quantum field theory
The calculation of the Meissel–Mertens constant
The third of Mertens' theorems*
Solution of the second kind to Bessel's equation
In the regularization/renormalization of the harmonic series as a finite value
The mean of the Gumbel distribution
The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
The answer to the coupon collector's problem*
In some formulations of Zipf's law
A definition of the cosine integral*
Lower bounds to a prime gap
An upper bound on Shannon entropy in quantum information theory Шаблон:R
Fisher–Orr model for genetics of adaptation in evolutionary biologyШаблон:R

Properties

The number Шаблон:Math has not been proved algebraic or transcendental. In fact, it is not even known whether Шаблон:Math is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if Шаблон:Math is rational, its denominator must be greater than 10²⁴⁴⁶⁶³.Шаблон:R The ubiquity of Шаблон:Math revealed by the large number of equations below makes the irrationality of Шаблон:Math a major open question in mathematics.Шаблон:R

However, some progress has been made. Kurt Mahler showed in 1968 that the number <math>\frac{\pi}{2}\frac{Y_0(2)}{J_0(2)}-\gamma</math> is transcendental (here, <math>J_\alpha(x)</math> and <math>Y_\alpha(x)</math> are Bessel functions).Шаблон:R Шаблон:R In 2009 Alexander Aptekarev proved that at least one of Euler's constant Шаблон:Math and the Euler–Gompertz constant Шаблон:Math is irrational;Шаблон:R Tanguy Rivoal proved in 2012 that at least one of them is transcendental.Шаблон:R Шаблон:R In 2010 M. Ram Murty and N. Saradha showed that at most one of the numbers of the form

<math display="block">\gamma(a,q) = \lim_{n\rightarrow\infty}\left(\left(\sum_{k=0}^n{\frac{1}{a+kq}}\right) - \frac{\log{(a+nq})}{q}\right)</math>

with Шаблон:Math and Шаблон:Math is algebraic; this family includes the special case Шаблон:Math.Шаблон:R Шаблон:Sfn In 2013 M. Ram Murty and A. Zaytseva found a different family containing Шаблон:Mvar, which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.Шаблон:R Шаблон:R

Relation to gamma function

Шаблон:Mvar is related to the digamma function Шаблон:Math, and hence the derivative of the gamma function Шаблон:Math, 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:Шаблон:R

<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 functions) 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

Шаблон:Mvar can also be expressed as an 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>

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 Шаблон:Mvar. 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:Шаблон:R

<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 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 Шаблон:Math are ceiling brackets. This formula indicates that when taking any positive integer Шаблон:Mvar and dividing it by each positive integer Шаблон:Mvar less than Шаблон:Mvar, the average fraction by which the quotient Шаблон:Math falls short of the next integer tends to Шаблон:Mvar (rather than 0.5) as Шаблон:Mvar 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 Шаблон:Math is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, Шаблон:Math. 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 Шаблон:Math

Шаблон:Mvar can also be expressed as follows where Шаблон:Mvar is the Glaisher–Kinkelin constant:

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

Шаблон:Mvar can also be expressed as follows, which can be proven by expressing the zeta function as a Laurent series:

<math display="block">\gamma=\lim_{n\to\infty}\left(-n+\zeta\Bigl(\frac{n+1}{n}\bigr)\right)</math>

Integrals

Шаблон:Mvar equals the value of a number of definite integrals:

<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 ,\\
&= \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 Шаблон:Math is the 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 stands for the Debye function value of Шаблон:Math, which is Шаблон:Math.

Definite integrals in which Шаблон:Mvar appears include:

<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} \end{align}</math>

One can express Шаблон:Mvar using a special case of Hadjicostas's formula as a double integral Шаблон:R 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 SondowШаблон:R 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 Шаблон:Math may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of seriesШаблон:R

<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 Шаблон:Math and Шаблон:Math are the number of 1s and 0s, respectively, in the base 2 expansion of Шаблон:Mvar.

We also have Catalan's 1875 integralШаблон:R

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

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 Шаблон:Math. However, the rate of convergence of this expansion depends significantly on Шаблон:Mvar. In particular, Шаблон:Math exhibits much more rapid convergence than the conventional expansion Шаблон:Math.Шаблон:R Шаблон:Sfn This is because

while

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

The series for Шаблон:Mvar is equivalent to a series Nielsen found in 1897:Шаблон:R Шаблон:Sfn

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

In 1910, Vacca found the closely related seriesШаблон:R

<math display="block">\begin{align} \gamma & = \sum_{k=2}^\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 Шаблон:Math is the logarithm to base 2 and Шаблон:Math is the floor function.

In 1926 he 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 Malmsten–Kummer expansion for the logarithm of the gamma functionШаблон:R 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>

An important expansion for Euler's constant is due to Fontana and 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 Шаблон:Math are Gregory coefficients.Шаблон:R This series is the special case Шаблон:Math 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 Шаблон:Math

A similar series with the Cauchy numbers of the second kind Шаблон:Math isШаблон:R

<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 Шаблон:Math 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 Шаблон:Mvar this series contains rational terms only. For example, at Шаблон:Math, it becomesШаблон:R

<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 Шаблон:Math is the gamma function.Шаблон:R

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 Шаблон:Math are the Gregory coefficients of the second order.Шаблон:R

As a series of prime numbers:

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

Asymptotic expansions

Шаблон:Mvar equals the following asymptotic formulas (where Шаблон:Math is the Шаблон:Mvarth 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> (Cesàro)

The third formula is also called the Ramanujan expansion.

Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations.Шаблон:R He showed that (Theorem A.1):

<math display="block">\begin{align} \sum_{n=1}^\infty \log n +\gamma - H_n + \frac{1}{2n} &= \frac{\log (2\pi)-1-\gamma}{2} \\ \sum_{n=1}^\infty \log \sqrt{n(n+1)} +\gamma - H_n &= \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 Шаблон:Math is important in number theory. Some authors denote this quantity simply as Шаблон:Math. Шаблон:Math equals the following limit, where Шаблон:Math is the Шаблон:Mvarth 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.Шаблон:R The numerical value of Шаблон:Math is:Шаблон:R

Шаблон:Block indent

Other infinite products relating to Шаблон:Math 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 Шаблон:Mvar-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 Шаблон:Mvarth factor is the Шаблон:Mathth 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 functions.Шаблон:R

It also holds thatШаблон:R

<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>

Continued fraction

The continued fraction expansion of Шаблон:Mvar begins Шаблон:Nowrap which has no apparent pattern. The continued fraction is known to have at least 475,006 terms,Шаблон:R and it has infinitely many terms if and only if Шаблон:Mvar is irrational.

Generalizations

Файл:Generalisation of Euler–Mascheroni constant.jpg

Шаблон:Math

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 Шаблон:Math, with Шаблон:Mvar as the special case Шаблон:Math.Шаблон:Sfn 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 Шаблон:Mvar. For example,

gives rise to the Stieltjes constants, and

gives

<math display="block">\gamma_{f_a} = \frac{(a-1)\zeta(a)-1}{a-1}</math>

where again the limit

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

appears.

A two-dimensional limit generalization is the Masser–Gramain constant.

Euler–Lehmer constants are given by summation of inverses of numbers in a common modulo class:Шаблон:Sfn

<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 Шаблон:Math then

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

Published digits

Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. 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.

Published decimal expansions of Шаблон:Mvar
Date	Decimal digits	Author	Sources
1734	5	Leonhard Euler
1735	15	Leonhard Euler
1781	16	Leonhard Euler
1790	32	Lorenzo Mascheroni, with 20–22 and 31–32 wrong
1809	22	Johann G. von Soldner
1811	22	Carl Friedrich Gauss
1812	40	Friedrich Bernhard Gottfried Nicolai
1857	34	Christian Fredrik Lindman
1861	41	Ludwig Oettinger
1867	49	William Shanks
1871	99	James W.L. Glaisher
1871	101	William Shanks
1877	262	J. C. Adams
1952	328	John William Wrench Jr.
1961	Шаблон:Val	Helmut Fischer and Karl Zeller
1962	Шаблон:Val	Donald Knuth	Шаблон:R
1962	Шаблон:Val	Dura W. Sweeney
1973	Шаблон:Val	William A. Beyer and Michael S. Waterman
1977	Шаблон:Val	Richard P. Brent
1980	Шаблон:Val	Richard P. Brent & Edwin M. McMillan
1993	Шаблон:Val	Jonathan Borwein
1999	Шаблон:Val	Patrick Demichel and Xavier Gourdon
March 13, 2009	Шаблон:Val	Alexander J. Yee & Raymond Chan	Шаблон:R
December 22, 2013	Шаблон:Val	Alexander J. Yee	Шаблон:R
March 15, 2016	Шаблон:Val	Peter Trueb	Шаблон:R
May 18, 2016	Шаблон:Val	Ron Watkins	Шаблон:R
August 23, 2017	Шаблон:Val	Ron Watkins	Шаблон:R
May 26, 2020	Шаблон:Val	Seungmin Kim & Ian Cutress	Шаблон:R
May 13, 2023	Шаблон:Val	Jordan Ranous & Kevin O'Brien	Шаблон:R
September 7, 2023	Шаблон:Val	Andrew Sun	Шаблон:R

References

Footnotes Шаблон:Reflist

External links

Шаблон:Springer
Шаблон:MathWorld
Jonathan Sondow.
Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
Further formulae which make use of the constant: Gourdon and Sebah (2004).

Шаблон:Leonhard Euler Шаблон:Authority control

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