Euler's constant

}</math>

| decimal= 0.57721...

thumb|237px|right|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 (), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by :

<math display="block">\begin{align}

\gamma &= \lim_{n\to\infty}\left(\sum_{k=1}^n \frac1{k}-\log n \right)\\

&=\int_1^\infty\left(\frac1{\lfloor x\rfloor}-\frac1x\right)\,\mathrm dx.

\end{align}</math>

Here, represents the floor function.

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

History

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Observations on harmonic progressions; Eneström Index 43), where he described it as "worthy of serious consideration". David Hilbert mentioned the irrationality of 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. Examples include, among others, the following places: (where <nowiki/>'*' means that this entry contains an explicit equation):

Analysis

The Weierstrass product formula for the gamma function and the Barnes G-function.
The asymptotic expansion of the gamma function, <math>\Gamma(1/x)\sim x-\gamma</math>.
Evaluations of the digamma function at rational values.
The Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants.
Values of the derivative of the Riemann zeta function and Dirichlet beta function.
In the regularization/renormalization of the harmonic series as a finite value.
Expressions involving the exponential and logarithmic integral.*
A definition of the cosine integral.*
Asymptotic expansions of modified Struve functions.
In relation to other special functions.

Number theory

An inequality for Euler's totient function.
The growth rate of the divisor function.
A formulation of the Riemann hypothesis.
The third of Mertens' theorems.*
Lower bounds to specific prime gaps.
An approximation of the average number of divisors of all numbers from 1 to a given n.
An estimation of the efficiency of the euclidean algorithm.
Sums involving the Möbius and von Mangolt function.
Estimate of the divisor summatory function of the Dirichlet hyperbola method.

In other fields

In some formulations of Zipf's law.
The answer to the coupon collector's problem.*
The mean of the Gumbel distribution.
An approximation of the Landau distribution.
The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
An upper bound on Shannon entropy in quantum information theory.
In dimensional regularization of Feynman diagrams in quantum field theory.
In the BCS equation on the critical temperature in BCS theory of superconductivity.*
Fisher–Orr model for genetics of adaptation in evolutionary biology.

Properties

Irrationality and transcendence

The number has not been proved algebraic or transcendental. In fact, it is not even known whether is irrational. The ubiquity of revealed by the large number of equations below and the fact that has been called the third most important mathematical constant after Pi| and E (mathematical constant)| makes the irrationality of a major open question in mathematics.

However, some progress has been made. In 1959 Andrei Shidlovsky proved that at least one of Euler's constant and the Gompertz constant is irrational; Tanguy Rivoal proved in 2012 that at least one of them is transcendental. 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. It is known that the transcendence degree of the field <math>\mathbb Q(e,\gamma,\delta)</math> is at least two.

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(\sum_{k=0}^n{\frac{1}{a+kq - \frac{\log{(a+nq})}{q} \right)</math>

is algebraic, if and ; this family includes the special case .

Using the same approach, in 2013, M. Ram Murty and A. Zaytseva showed that the generalized Euler constants have the same property,

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 is a fixed list of prime numbers, <math>1_\Omega(n) =0</math> if at least one of the primes in is a prime factor of , and <math>1_\Omega(n) =1</math> otherwise. In particular, .

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

Euler's constant is conjectured not to be an algebraic period, but the values of its first 109 decimal digits seem to indicate that it could be a normal number.

Continued fraction

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

:<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, and it has infinitely many terms if and only if is irrational.

[[File:KhinchinBeispiele.svg|thumb|The Khinchin limits for <math>\pi</math> (red), <math>\gamma</math> (blue) and <math>\sqrt[3]{2}</math> (green).|350x350px]]

Numerical evidence suggests that both Euler's constant as well as the constant 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. However neither of these limits has been proven.

There also exists a generalized continued fraction for Euler's constant.

A good simple approximation of is given by the reciprocal of the square root of 3 or about 0.57735:

:<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 identities

Relation to gamma function

is related to the digamma function (not to be confused with wave function), and hence the derivative of the gamma function , 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:

<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

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:

:<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 . 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:

<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 are ceiling brackets.

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

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

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

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 numbers

Numerous formulations have been derived that express <math>\gamma</math> in terms of sums and logarithms of triangular numbers. One of the earliest of these is a formula for the harmonic number attributed to Srinivasa Ramanujan where <math>\gamma</math> is related to <math>\textstyle \log 2T_{k}</math> in a series that considers the powers of <math>\textstyle \frac{1}{T_{k</math> (an earlier, less-generalizable proof by Ernesto Cesàro gives the first two terms of the series, with an error term):

:<math>\begin{align}

\gamma

&= H_u - \frac{1}{2} \log 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 follows a similar series:

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

Integrals

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 \\

&= \frac{1}{2}+\int_{0}^{\infty}\frac{2x\,dx}{(x^2+1)(e^{2\pi x}-1)} \\

&= \frac{1}{\pi}\int_{0}^{\pi}\frac{\sin x}{x}e^{x\cot x}\log\left(\frac{\sin x}{x}e^{x\cot x}\right)dx

\end{align}

</math>

where 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 .

Definite integrals in which 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}

\\ \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 integral

<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 using a special case of Hadjicostas's formula as a double integral 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 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 may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series

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

Series expansions

In general,

\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 . However, the rate of convergence of this expansion depends significantly on . In particular, exhibits much more rapid convergence than the conventional expansion . This is because

\frac{1}{2(n+1)} < \gamma_n(0) - \gamma < \frac{1}{2n},

</math>

while

\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 :

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

The series for is equivalent to a series Nielsen found in 1897:

<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 series

<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 is the logarithm to base 2 and is the floor function.

This can be generalized to:

<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

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

Other infinite products relating to 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 -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 th factor is the 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 functions.

It also holds that

<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 digits

{| class="wikitable" style="margin: 1em auto 1em auto"

|+ Published decimal expansions of

! Date || Decimal digits || Author || Sources

| 1734 || style="text-align:right;"| 5 || Leonhard Euler ||

| 1735 || style="text-align:right;"| 15 || Leonhard Euler ||

| 1781 || style="text-align:right;"| 16 || Leonhard Euler ||

| 1790 || style="text-align:right;"| 32 || Lorenzo Mascheroni, with 20–22 and 31–32 wrong ||

| 1809 || style="text-align:right;"| 22 || Johann G. von Soldner ||

| 1811 || style="text-align:right;"| 22 || Carl Friedrich Gauss ||

| 1812 || style="text-align:right;"| 40 || Friedrich Bernhard Gottfried Nicolai ||

| 1861 || style="text-align:right;" | 41 || Ludwig Oettinger ||

| 1867 || style="text-align:right;" | 49 || William Shanks||

| 1871 || style="text-align:right;" | 100 || James W.L. Glaisher||

| 1877 || style="text-align:right;" | 263 || J. C. Adams||

| 1952 || style="text-align:right;" | 328 || John William Wrench Jr.||

| 1961 || style="text-align:right;" | || Helmut Fischer and Karl Zeller||

| 1962 || style="text-align:right;"| || Donald Knuth ||

| 1963 || style="text-align:right;" | || Dura W. Sweeney ||

| 1973 || style="text-align:right;"| || William A. Beyer and Michael S. Waterman ||

| 1977 || style="text-align:right;"| || Richard P. Brent ||

| 1993 || style="text-align:right;"| || Jonathan Borwein ||

| 1997 || style="text-align:right;" | || Thomas Papanikolaou ||

:<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>.

References

Footnotes

External links

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

Euler's constant

History

Analysis

Number theory

In other fields

Properties

Irrationality and transcendence

Continued fraction

Formulas and identities

Relation to gamma function

Relation to the zeta function

Relation to triangular numbers

Integrals

Series expansions

Published digits

See also

References

Footnotes

Further reading

External links