Euler 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 :

\begin
\gamma &= \lim_\left(-\log n + \sum_^n \frac1\right)\\ px&=\int_1^\infty\left(-\frac1x+\frac1\right)\,\mathrm dx.
\end

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'' (Eneström Index 43), where he described it as "worthy of serious consideration". Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations and 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 and for the constant. Other computations were done by Johann von Soldner in 1809, who used the notation . The notation 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. For example, the German mathematician Carl Anton Bretschneider used the notation in 1835, and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842. Euler's constant was also studied by the Indian mathematician Srinivasa Ramanujan who published one paper on it in 1917. 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.

Appearances

Euler's constant appears frequently in mathematics, especially in number theory and analysis. Examples include, among others, the following places: (''where'' '''*' 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, $\Gamma(1/x)\sim x-\gamma$.
* 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 connection to the Laplace and Mellin transform.
* 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.*
* In relation to Bessel functions.
* 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.*
* The calculation of the Meissel–Mertens constant.
* Lower bounds to specific prime gaps.
* An approximation of the average number of divisors of all numbers from 1 to a given ''n.''
* The Lenstra–Pomerance–Wagstaff conjecture on the frequency of Mersenne primes.
* 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 and 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 $\frac \pi 2\frac-\gamma$ is transcendental, where $J_0$ and $Y_0$ are the usual Bessel functions. It is known that the transcendence degree of the field $\mathbb Q(e,\gamma,\delta)$ 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
$\gamma(a,q) = \lim_\left( - \frac + \sum_^n\right)$
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
$\gamma(\Omega) = \lim_ \left( \sum_^x \frac - \log x \cdot \lim_ \frac \right),$
where is a fixed list of prime numbers, $1_\Omega(n) =0$ if at least one of the primes in is a prime factor of , and $1_\Omega(n) =1$ 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:

:$\gamma=0+\cfrac$

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.

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 $p_n/q_n$ are the convergents of their respective continued fractions, the limit $\lim_q_n^$ 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:

:$\frac1\sqrt =0+\cfrac$

with the difference being about 1 in 7,429.

Formulas and identities

Relation to gamma function

is related to the digamma function , and hence the derivative of the gamma function , when both functions are evaluated at 1. Thus:

$-\gamma = \Gamma'(1) = \Psi(1).$

This is equal to the limits:

$\begin-\gamma &= \lim_\left(\Gamma(z) - \frac1\right) \\&= \lim_\left(\Psi(z) + \frac1\right).\end$

Further limit results are:

$\begin \lim_\frac1\left(\frac1 - \frac1\right) &= 2\gamma \\ \lim_\frac1\left(\frac1 - \frac1\right) &= \frac. \end$

A limit related to the beta function (expressed in terms of gamma functions) is

$\begin \gamma &= \lim_\left(\frac - \frac\right) \\ &= \lim\limits_\sum_^m\frac\log\big(\Gamma(k+1)\big). \end$

Relation to the zeta function

can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:

$\begin\gamma &= \sum_^ (-1)^m\frac \\ &= \log\frac4 + \sum_^ (-1)^m\frac.\end$
The constant $\gamma$ can also be expressed in terms of the sum of the reciprocals of non-trivial zeros $\rho$ of the zeta function: 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.
:$\gamma = \log 4\pi + \sum_ \frac - 2$
Other series related to the zeta function include:

$\begin \gamma &= \tfrac3- \log 2 - \sum_^\infty (-1)^m\,\frac\big(\zeta(m)-1\big) \\ &= \lim_\left(\frac - \log n + \sum_^n \left(\frac1 - \frac\right)\right) \\ &= \lim_\left(\frac \sum_^\infty \frac \sum_^m \frac1 - n \log 2+ O \left (\frac1\right)\right).\end$

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:

$\begin \gamma &= \lim_\sum_^\infty \left(\frac1-\frac1\right) \\&= \lim_\left(\zeta(s) - \frac\right) \\&= \lim_\frac \end$

and the following formula, established in 1898 by de la Vallée-Poussin:

$\gamma = \lim_\frac1\, \sum_^n \left(\left\lceil \frac \right\rceil - \frac\right)$

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:

$\gamma =\lim_\left( \sum_^n \frac1 - \log n -\sum_^\infty \frac\right),$

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:

$H_n = \log(n) + \gamma + \frac1 - \frac1 + \frac1 - \varepsilon,$
where

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

$\gamma =12\,\log(A)-\log(2\pi)+\frac\,\zeta'(2)$

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

$\gamma=\lim_\left(-n+\zeta\left(\frac\right)\right)$

Relation to triangular numbers

Numerous formulations have been derived that express $\gamma$ in terms of sums and logarithms of triangular numbers. See formulas 1 and 10. One of the earliest of these is a formula for the harmonic number attributed to Srinivasa Ramanujan where $\gamma$ is related to $\textstyle \ln 2T_$ in a series that considers the powers of $\textstyle \frac$ (an earlier, less-generalizable proof by Ernesto Cesàro gives the first two terms of the series, with an error term):

:$\begin \gamma &= H_u - \frac \ln 2T_u - \sum_^\frac-\Theta_\,\frac \end$

From Stirling's approximation follows a similar series:

:$\gamma = \ln 2\pi - \sum_^ \frac$

The series of inverse triangular numbers also features in the study of the Basel problem posed by Pietro Mengoli. Mengoli proved that $\textstyle \sum_^\infty \frac = 1$, a result Jacob Bernoulli later used to estimate the value of $\zeta(2)$, placing it between $1$ and $\textstyle \sum_^\infty \frac = \sum_^\infty \frac = 2$. This identity appears in a formula used by Bernhard Riemann to compute roots of the zeta function, where $\gamma$ is expressed in terms of the sum of roots $\rho$ plus the difference between Boya's expansion and the series of exact unit fractions $\textstyle \sum_^ \frac$:

:$\gamma - \ln 2 = \ln 2\pi + \sum_ \frac - \sum_^ \frac$

Integrals

equals the value of a number of definite integrals:

$\begin \gamma &= - \int_0^\infty e^ \log x \,dx \\ &= -\int_0^1\log\left(\log\frac 1 x \right) dx \\ &= \int_0^\infty \left(\frac1-\frac1 \right)dx \\ &= \int_0^1\frac \, dx -\int_1^\infty \frac\, dx\\ &= \int_0^1\left(\frac1 + \frac1\right)dx\\ &= \int_0^\infty \left(\frac1-e^\right)\frac,\quad k>0\\ &= 2\int_0^\infty \frac \, dx ,\\ &= \log\frac-\int_0^\infty \frac \, dx ,\\ &= \int_0^1 H_x \, dx, \\ &= \frac+\int_^\log\left(1+\frac\right)dt \\ &= 1-\int_0^1 \ dx \\ &= \frac+\int_^\frac \end$
where is the fractional harmonic number, and $\$ is the fractional part of $1/x$.

The third formula in the integral list can be proved in the following way:

\begin
&\int_0^ \left(\frac - \frac \right) dx
= \int_0^ \frac dx
= \int_0^ \frac \sum_^ \frac dx \\ pt&= \int_0^ \sum_^ \frac dx
= \sum_^ \int_0^ \frac dx
= \sum_^ \frac \int_0^ \frac dx \\ pt&= \sum_^ \frac m!\zeta(m+1)
= \sum_^ \frac\zeta(m+1)
= \sum_^ \frac \sum_^\frac
= \sum_^ \sum_^ \frac\frac \\ pt&= \sum_^ \sum_^ \frac\frac
= \sum_^ \left frac - \log\left(1+\frac\right)\right = \gamma
\end

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:

$\begin \int_0^\infty e^ \log x \,dx &= -\frac \\ \int_0^\infty e^ \log^2 x \,dx &= \gamma^2 + \frac \\ \int_0^\infty \frac \,dx &= \frac12 \log^2 2 - \gamma \end$

We also have Catalan's 1875 integral

$\gamma = \int_0^1 \left(\frac1\sum_^\infty x^\right)\,dx.$

One can express using a special case of Hadjicostas's formula as a double integral with equivalent series:

$\begin \gamma &= \int_0^1 \int_0^1 \frac\,dx\,dy \\ &= \sum_^\infty \left(\frac 1 n -\log\frac n \right). \end$

An interesting comparison by Sondow is the double integral and alternating series

$\begin \log\frac 4 \pi &= \int_0^1 \int_0^1 \frac \,dx\,dy \\ &= \sum_^\infty \left((-1)^\left(\frac 1 n -\log\frac n \right)\right). \end$

It shows that may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series

$\begin \gamma &= \sum_^\infty \frac \\ \log\frac4 &= \sum_^\infty \frac , \end$

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

Series expansions

In general,

$\gamma = \lim_\left(\frac+\frac+\frac + \ldots + \frac - \log(n+\alpha) \right) \equiv \lim_ \gamma_n(\alpha)$

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

$\frac < \gamma_n(0) - \gamma < \frac,$

while

$\frac < \gamma_n(1/2) - \gamma < \frac.$

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 :
$\gamma = \sum_^\infty \left(\frac 1 k - \log\left(1+\frac 1 k \right)\right).$

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

$\gamma = 1 - \sum_^\infty (-1)^k\frac.$

In 1910, Vacca found the closely related series

\begin
\gamma & = \sum_^\infty (-1)^k\frac k \\ pt& = \tfrac12-\tfrac13 + 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right) + 3\left(\tfrac18 - \tfrac19 + \tfrac1 - \tfrac1 + \cdots - \tfrac1\right) + \cdots,
\end

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

This can be generalized to:

$\gamma= \sum_^\infty \frac \varepsilon(k)$where:$\varepsilon(k)= \begin B-1, &\text B\mid n \\ -1, &\textB\nmid n \end$

In 1926 Vacca found a second series:

\begin
\gamma + \zeta(2) & = \sum_^\infty \left( \frac1 - \frac1\right) \\ pt& = \sum_^\infty \frac \\ pt&= \frac12 + \frac23 + \frac1\sum_^ \frac + \frac1\sum_^ \frac + \cdots
\end

From the Malmsten– Kummer expansion for the logarithm of the gamma function we get:

$\gamma = \log\pi - 4\log\left(\Gamma(\tfrac34)\right) + \frac 4 \pi \sum_^\infty (-1)^\frac.$

Ramanujan, in his lost notebook gave a series that approaches :

$\gamma = \log 2 - \sum_^ \sum_^ \frac$

An important expansion for Euler's constant is due to Fontana and Mascheroni

$\gamma = \sum_^\infty \frac = \frac12 + \frac1 + \frac1 + \frac + \frac3 + \cdots,$
where are Gregory coefficients. This series is the special case of the expansions

$\begin \gamma &= H_ - \log k + \sum_^\frac && \\ &= H_ - \log k + \frac1 + \frac1 + \frac1 + \frac + \cdots && \end$

convergent for

A similar series with the Cauchy numbers of the second kind is

$\gamma = 1 - \sum_^\infty \frac =1- \frac -\frac - \frac - \frac - \frac - \ldots$

Blagouchine (2018) found a generalisation of the Fontana–Mascheroni series

$\gamma=\sum_^\infty\frac\Big\, \quad a>-1$

where are the Bernoulli polynomials of the second kind, which are defined by the generating function

$\frac= \sum_^\infty z^n \psi_n(s) ,\qquad , z, <1.$

For any rational this series contains rational terms only. For example, at , it becomes

$\gamma=\frac - \frac - \frac - \frac - \frac - \frac - \frac - \ldots$
Other series with the same polynomials include these examples:

$\gamma= -\log(a+1) - \sum_^\infty\frac,\qquad \Re(a)>-1$

and

$\gamma= -\frac \left\,\qquad \Re(a)>-1$

where is the gamma function.

A series related to the Akiyama–Tanigawa algorithm is

$\gamma= \log(2\pi) - 2 - 2 \sum_^\infty\frac= \log(2\pi) - 2 + \frac + \frac+ \frac + \frac+ \frac + \ldots$

where are the Gregory coefficients of the second order.

As a series of prime numbers:

$\gamma = \lim_\left(\log n - \sum_\frac\right).$

Asymptotic expansions

equals the following asymptotic formulas (where is the th harmonic number):

*$\gamma \sim H_n - \log n - \frac1 + \frac1 - \frac1 + \cdots$ (''Euler'')
*$\gamma \sim H_n - \log\left(\right)$ (''Negoi'')
*$\gamma \sim H_n - \frac - \frac1 + \frac1 - \cdots$ ('' Cesàro'')

The third formula is also called the Ramanujan expansion.

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

$\begin \sum_^\infty \Big(\log n +\gamma - H_n + \frac\Big) &= \frac \\ \sum_^\infty \Big(\log \sqrt +\gamma - H_n \Big) &= \frac-\gamma \\ \sum_^\infty (-1)^n\Big(\log n +\gamma - H_n\Big) &= \frac \end$

Exponential

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

equals the following limit, where is the th prime number:

$e^\gamma = \lim_\frac1 \prod_^n \frac.$

This restates the third of Mertens' theorems.

We further have the following product involving the three constants , and :

$\frac=\lim_ \log p_n \prod_^n \frac.$

Other infinite products relating to include:

$\begin \frac &= \prod_^\infty e^\left(1+\frac1\right)^n \\ \frac &= \prod_^\infty e^\left(1+\frac2\right)^n. \end$

These products result from the Barnes -function.

In addition,

e^ = \sqrt \cdot \sqrt \cdot \sqrt \cdot \sqrt \cdots

where the th factor is the th root of

$\prod_^n (k+1)^.$

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using hypergeometric functions.

It also holds that

$\frac=\prod_^\infty\left(e^\left(1+\frac+\frac\right)\right).$

Published digits

Generalizations

Stieltjes constants

''Euler's generalized constants'' are given by

$\gamma_\alpha = \lim_\left(\sum_^n \frac1 - \int_1^n \frac1\,dx\right)$

for , with as the special case . Extending for gives:

$\gamma_ = \zeta(\alpha) - \frac1$

with again the limit:

$\gamma = \lim_\left(\zeta(a) - \frac1\right)$

This can be further generalized to

$c_f = \lim_\left(\sum_^n f(k) - \int_1^n f(x)\,dx\right)$

for some arbitrary decreasing function . Setting

$f_n(x) = \frac$

gives rise to the Stieltjes constants $\gamma_n$, that occur in the Laurent series expansion of the Riemann zeta function:

: $\zeta(1+s)=\frac+\sum_^\infty \frac \gamma_n s^n.$

with $\gamma_0 = \gamma = 0.577\dots$

Euler-Lehmer constants

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

$\gamma(a,q) = \lim_\left (\sum_ \frac1-\frac\right).$

The basic properties are

$\begin &\gamma(0,q) = \frac, \\ &\sum_^ \gamma(a,q)=\gamma, \\ &q\gamma(a,q) = \gamma-\sum_^e^\log\left(1-e^\right), \end$

and if the greatest common divisor then

$q\gamma(a,q) = \frac\gamma\left(\frac,\frac\right)-\log d.$

Masser-Gramain constant

A two-dimensional generalization of Euler's constant is the Masser-Gramain constant. It is defined as the following limiting difference:

:$\delta = \lim_ \left( -\log n + \sum_^n \frac \right)$

where $r_k$ is the smallest radius of a disk in the complex plane containing at least $k$ Gaussian integers.

The following bounds have been established: $1.819776 < \delta < 1.819833$.

See also

* Harmonic series
* Riemann zeta function
* Stieltjes constants
* Meissel-Mertens constant

References

*
*
*

Footnotes

Further reading

* Derives as sums over Riemann zeta functions.
*
*
*
*
*
* Julian Havil (2003): ''GAMMA: Exploring Euler's Constant'', Princeton University Press, ISBN 978-0-69114133-6.
*
*
*
*
*
*
* with an Appendix b
Sergey Zlobin

External links

*
*

Jonathan Sondow.


E.A. Karatsuba (2005)
* Further formulae which make use of the constant

{{DEFAULTSORT:Euler's constant
Mathematical constants
Unsolved problems in number theory
Leonhard Euler