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Euler's constant

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Template:Short description Template:Redirect Template:Distinguish Template:Use shortened footnotes Template:Log(x) Template:Infobox mathematical constant

File:Gamma-area.svg
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 (Template:Math), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by Template: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)\,\mathrm dx. \end{align}</math>

Here, Template:Math represents the floor function.

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

Template:Block indent

HistoryEdit

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".<ref name=":3" />Template:Sfn Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations Template:Math and Template:Math 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 Template:Math and Template:Math for the constant. Other computations were done by Johann von Soldner in 1809, who used the notation Template:Math. The notation Template: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.Template:Sfn For example, the German mathematician Carl Anton Bretschneider used the notation Template:Math in 1835,Template:Sfn and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.Template:R Euler's constant was also studied by the Indian mathematician Srinivasa Ramanujan who published one paper on it in 1917.<ref>Template:Cite book</ref> David Hilbert mentioned the irrationality of Template:Math 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.<ref name=":3" />

AppearancesEdit

Euler's constant appears frequently in mathematics, especially in number theory and analysis.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Examples include, among others, the following places: (where '*' means that this entry contains an explicit equation):

AnalysisEdit

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Number theoryEdit

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In other fieldsEdit

PropertiesEdit

Irrationality and transcendenceEdit

The number Template:Math has not been proved algebraic or transcendental. In fact, it is not even known whether Template:Math is irrational. The ubiquity of Template:Math revealed by the large number of equations below and the fact that Template:Math has been called the third most important mathematical constant after [[Pi|Template:Math]] and [[E (mathematical constant)|Template:Math]]<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=":7">Template:Cite book</ref> makes the irrationality of Template:Math a major open question in mathematics.<ref name=":3" /><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>Template:R<ref name=":6">Template:Cite book</ref>

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However, some progress has been made. In 1959 Andrei Shidlovsky proved that at least one of Euler's constant Template:Math and the Gompertz constant Template:Math is irrational;Template:R<ref name=":2">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Tanguy Rivoal proved in 2012 that at least one of them is transcendental.Template:R Kurt Mahler showed in 1968 that the number <math display=inline>\frac \pi 2\frac{Y_0(2)}{J_0(2)}-\gamma</math> is transcendental, where <math>J_0</math> and <math>Y_0</math> are the usual Bessel functions.Template:RTemplate:Sfn It is known that the transcendence degree of the field <math>\mathbb Q(e,\gamma,\delta)</math> is at least two.Template:Sfn

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 <math display="block">\gamma(a,q) = \lim_{n\rightarrow\infty}\left( - \frac{\log{(a+nq})}{q} + \sum_{k=0}^n{\frac{1}{a+kq}}\right)</math> is algebraic, if Template:Math and Template:Math; this family includes the special case Template:Math.Template:SfnTemplate:R

Using the same approach, in 2013, M. Ram Murty and A. Zaytseva showed that the generalized Euler constants have the same property, Template:SfnTemplate:R <ref>Template:Cite journal</ref> where the generalized Euler constant are defined as <math display="block">\gamma(\Omega) = \lim_{x\rightarrow\infty} \left( \sum_{n=1}^x \frac{1_\Omega(n)}{n} - \log x \cdot \lim_{x\rightarrow\infty} \frac{ \sum_{n=1}^x 1_\Omega (n) }{x} \right),</math> where Template:Tmath is a fixed list of prime numbers, <math>1_\Omega(n) =0</math> if at least one of the primes in Template:Tmath is a prime factor of Template:Tmath, and <math>1_\Omega(n) =1</math> otherwise. In particular, Template:Tmath.

Using a continued fraction analysis, Papanikolaou showed in 1997 that if Template:Math is rational, its denominator must be greater than 10244663.Template:R If Template:Math is a rational number, then its denominator must be greater than 1015000.Template:Sfn

Euler's constant is conjectured not to be an algebraic period,Template:Sfn but the values of its first 109 decimal digits seem to indicate that it could be a normal number.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Continued fractionEdit

The simple continued fraction expansion of Euler's constant is given by:Template:R

<math>\gamma=0+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{4+\dots}}}}}}}</math>

which has no apparent pattern. It is known to have at least 16,695,000,000 terms,Template:R and it has infinitely many terms if and only if Template:Mvar is irrational.

File:KhinchinBeispiele.svg
The Khinchin limits for <math>\pi</math> (red), <math>\gamma</math> (blue) and <math>\sqrt[3]{2}</math> (green).

Numerical evidence suggests that both Euler's constant Template:Math as well as the constant Template:Math are among the numbers for which the geometric mean of their simple continued fraction terms converges to Khinchin's constant. Similarly, when <math>p_n/q_n</math> are the convergents of their respective continued fractions, the limit <math>\lim_{n\to\infty}q_n^{1/n}</math> appears to converge to Lévy's constant in both cases.<ref name=":4">Template:Cite journal</ref> However neither of these limits has been proven.<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

There also exists a generalized continued fraction for Euler's constant.<ref>Template:Citation</ref>

A good simple approximation of Template:Math is given by the reciprocal of the square root of 3 or about 0.57735:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>\frac1\sqrt {3}=0+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\dots}}}}}}}</math>

with the difference being about 1 in 7,429.

Formulas and identitiesEdit

Relation to gamma functionEdit

Template:Mvar is related to the digamma function Template:Math, and hence the derivative of the gamma function Template: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:Template: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 functionEdit

Template: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>

The constant <math>\gamma</math> can also be expressed in terms of the sum of the reciprocals of non-trivial zeros <math>\rho</math> of the zeta function:<ref name="Marek6infinity2019">Template:Cite arXiv 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 Template: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:Template: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 Template:Math are ceiling brackets. This formula indicates that when taking any positive integer Template:Mvar and dividing it by each positive integer Template:Mvar less than Template:Mvar, the average fraction by which the quotient Template:Math falls short of the next integer tends to Template:Mvar (rather than 0.5) as Template: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 Template:Math is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, Template: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 Template:Math

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

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

Template: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\left(\frac{n+1}{n}\right)\right)</math>

Relation to triangular numbersEdit

Numerous formulations have been derived that express <math>\gamma</math> in terms of sums and logarithms of triangular numbers.<ref name="Boya2008AnotherRelation">Template:Cite journal See formulas 1 and 10.</ref><ref name="Jonathan2005DoubleIntegrals">Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> One of the earliest of these is a formula<ref>Template:Cite arXiv See formula 1.8 on page 3.</ref><ref>Template:Cite journal</ref> for the Template:Nowrap 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>Template:Cite journal</ref><ref>Template:Cite book 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>Template:Cite book 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 problemTemplate:Sfn<ref>Template:Cite journal</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 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 roots of the zeta function,<ref>Template:Cite book</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 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>

IntegralsEdit

Template: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 ,\\

&= \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 Template: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 is the definition of the Riemann zeta function, which is Template:Math.

Definite integrals in which Template:Mvar appears include:<ref name=":3">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=":1">Template:Cite journal</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 Catalan's 1875 integralTemplate:R

<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 Template:Mvar using a special case of Hadjicostas's formula as a double integralTemplate: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 SondowTemplate: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 Template:Math may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of seriesTemplate: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 Template:Math and Template:Math are the number of 1s and 0s, respectively, in the base 2 expansion of Template:Mvar.

Series expansionsEdit

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 Template:Math. However, the rate of convergence of this expansion depends significantly on Template:Mvar. In particular, Template:Math exhibits much more rapid convergence than the conventional expansion Template:Math.Template:RTemplate:Sfn 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 Template: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 Template:Mvar is equivalent to a series Nielsen found in 1897:Template:RTemplate: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 seriesTemplate:R

<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 Template:Math is the logarithm to base 2 and Template:Math is the floor function.

This can be generalized to:<ref>Template:Citation</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 Malmsten–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 lost notebook gave a series that approaches Template:MvarTemplate:R:

<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 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 Template:Math are Gregory coefficients.Template:R This series is the special case Template: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 Template:Math

A similar series with the Cauchy numbers of the second kind Template:Math isTemplate: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 a 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 Template: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 Template:Mvar this series contains rational terms only. For example, at Template:Math, it becomesTemplate: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 Template:Math is the gamma function.Template: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 Template:Math are the Gregory coefficients of the second order.Template: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 expansionsEdit

Template:Mvar equals the following asymptotic formulas (where Template:Math is the Template: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.Template:R 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>

ExponentialEdit

The constant Template:Math is important in number theory. Its numerical value is:Template:R

Template:Block indent Template:Math equals the following limit, where Template:Math is the Template: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.Template:R

We further have the following product involving the three constants Template:Math, Template:Math and Template:Math:<ref name=":9">{{#invoke:citation/CS1|citation |CitationClass=web }}</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 products relating to Template: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 Template: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 Template:Mvarth factor is the Template: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.Template:R

It also holds thatTemplate: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>

Published digitsEdit

Published decimal expansions of Template:Mvar
Date Decimal digits Author Sources
1734 5 Leonhard Euler Template:Sfn
1735 15 Leonhard Euler Template:Sfn
1781 16 Leonhard Euler Template:Sfn
1790 32 Lorenzo Mascheroni, with 20–22 and 31–32 wrong Template:Sfn
1809 22 Johann G. von Soldner Template:Sfn
1811 22 Carl Friedrich Gauss Template:Sfn
1812 40 Friedrich Bernhard Gottfried Nicolai Template:Sfn
1861 41 Ludwig Oettinger <ref>Template:Cite journal</ref>
1867 49 William Shanks <ref>Template:Cite journal</ref>
1871 100 James W.L. Glaisher Template:Sfn
1877 263 J. C. Adams Template:Sfn
1952 328 John William Wrench Jr. Template:Sfn
1961 Template:Val Helmut Fischer and Karl Zeller citation CitationClass=web

}}</ref>

1962 Template:Val Donald Knuth Template:R
1963 Template:Val Dura W. Sweeney <ref>Template:Cite journal</ref>
1973 Template:Val William A. Beyer and Michael S. Waterman <ref>Template:Cite journal</ref>
1977 Template:Val Richard P. Brent <ref name=":4" />
1980 Template:Val Richard P. Brent & Edwin M. McMillan <ref>Template:Cite journal</ref>
1993 Template:Val Jonathan Borwein citation CitationClass=web

}}</ref>

1997 Template:Val Thomas Papanikolaou citation CitationClass=web

}}</ref>

1998 Template:Val Xavier Gourdon citation CitationClass=web

}}</ref>

1999 Template:Val Patrick Demichel and Xavier Gourdon <ref name=":5" />
March 13, 2009 Template:Val Alexander J. Yee & Raymond Chan Template:R
December 22, 2013 Template:Val Alexander J. Yee Template:R
March 15, 2016 Template:Val Peter Trueb Template:R
May 18, 2016 Template:Val Ron Watkins Template:R
August 23, 2017 Template:Val Ron Watkins Template:R
May 26, 2020 Template:Val Seungmin Kim & Ian Cutress Template:R
May 13, 2023 Template:Val Jordan Ranous & Kevin O'Brien Template:R
September 7, 2023 Template:Val Andrew Sun Template:R

GeneralizationsEdit

Stieltjes constantsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

File:Generalisation of Euler–Mascheroni constant.jpg
Euler's generalized constants Template:Math for Template: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 Template:Math, with Template:Mvar as the special case Template:Math.Template:Sfn Extending for Template:Math 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 Template:Mvar. 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>

n approximate value of γn OEIS
0 +0.5772156649015 A001620
1 −0.0728158454836 A082633
2 −0.0096903631928 A086279
3 +0.0020538344203 A086280
4 +0.0023253700654 A086281
100 −4.2534015717080 × 1017
1000 −1.5709538442047 × 10486

Euler-Lehmer constantsEdit

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

<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 Template:Math 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 constantEdit

A two-dimensional generalization of Euler's constant is the Masser-Gramain constant. It is defined as the following limiting difference:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 integers.

The following bounds have been established: <math>1.819776 < \delta < 1.819833</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

See alsoEdit

ReferencesEdit

FootnotesEdit

Template:Reflist

Further readingEdit

|CitationClass=web }}

  • {{#invoke:citation/CS1|citation

|CitationClass=web }}

External linksEdit

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Euler-MascheroniConstant%7CEuler-MascheroniConstant.html}} |title = Euler–Mascheroni constant |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = none }}

Template:Leonhard Euler Template:Authority control

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