Euler's constant

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Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:$${\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)\\[5px]&=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,\mathrm {d} x.\end{aligned}}}$$Here, ⌊·⌋ represents the floor function.

The numerical value of Euler's constant, to 50 decimal places, is:^[1]

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".^[2][3] Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations C and O 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 A and a for the constant. Other computations were done by Johann von Soldner in 1809, who used the notation H. The notation γ 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.^[3] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835,^[4] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.^[5] Euler's constant was also studied by the Indian mathematician Srinivasa Ramanujan who published one paper on it in 1917.^[6] David Hilbert mentioned the irrationality of γ 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.^[2]

Euler's constant appears frequently in mathematics, especially in number theory and analysis.^[7] Examples include, among others, the following places: (where '*' means that this entry contains an explicit equation):

• The Weierstrass product formula for the gamma function and the Barnes G-function.^[8][9]
• The asymptotic expansion of the gamma function, ${\displaystyle \Gamma (1/x)\sim x-\gamma }$.
• Evaluations of the digamma function at rational values.^[10]
• The Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants.^[11]
• Values of the derivative of the Riemann zeta function and Dirichlet beta function.^{[12]: 137 [13]}
• In connection to the Laplace and Mellin transform.^[14][15]
• In the regularization/renormalization of the harmonic series as a finite value.
• Expressions involving the exponential and logarithmic integral.*^[16][17]
• A definition of the cosine integral.*^[16]
• In relation to Bessel functions.^{[18][19][20][21]}
• Asymptotic expansions of modified Struve functions.^[22]
• In relation to other special functions.^[23][24]

• An inequality for Euler's totient function.^[25]
• The growth rate of the divisor function.^[26]
• A formulation of the Riemann hypothesis.^[27][28]
• The third of Mertens' theorems.*^[29]
• The calculation of the Meissel–Mertens constant.^[30]
• Lower bounds to specific prime gaps.^[31]
• An approximation of the average number of divisors of all numbers from 1 to a given n.^[32]
• The Lenstra–Pomerance–Wagstaff conjecture on the frequency of Mersenne primes.^[33]
• An estimation of the efficiency of the euclidean algorithm.^[34]
• Sums involving the Möbius and von Mangolt function.
• Estimate of the divisor summatory function of the Dirichlet hyperbola method.^[35]

• 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.^[36]
• 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.^[37]

The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. The ubiquity of γ revealed by the large number of equations below and the fact that γ has been called the third most important mathematical constant after π and e^[38][12] makes the irrationality of γ a major open question in mathematics.^{[2][39][40][32]}

However, some progress has been made. In 1959 Andrei Shidlovsky proved that at least one of Euler's constant γ and the Gompertz constant δ is irrational;^[41][27] Tanguy Rivoal proved in 2012 that at least one of them is transcendental.^[42] Kurt Mahler showed in 1968 that the number ${\textstyle {\frac {\pi }{2}}{\frac {Y_{0}(2)}{J_{0}(2)}}-\gamma }$ is transcendental, where ${\displaystyle J_{0}}$ and ${\displaystyle Y_{0}}$ are the usual Bessel functions.^[43][3] It is known that the transcendence degree of the field ${\displaystyle \mathbb {Q} (e,\gamma ,\delta )}$ is at least two.^[3]

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 $${\displaystyle \gamma (a,q)=\lim _{n\rightarrow \infty }\left(-{\frac {\log {(a+nq})}{q}}+\sum _{k=0}^{n}{\frac {1}{a+kq}}\right)}$$ is algebraic, if q ≥ 2 and 1 ≤ a < q; this family includes the special case γ(2,4) = γ/4.^[3][44]

Using the same approach, in 2013, M. Ram Murty and A. Zaytseva showed that the generalized Euler constants have the same property, ^{[3][45] [46]} where the generalized Euler constant are defined as $${\displaystyle \gamma (\Omega )=\lim _{x\rightarrow \infty }\left(\sum _{n=1}^{x}{\frac {1_{\Omega }(n)}{n}}-\log x\cdot \lim _{x\rightarrow \infty }{\frac {\sum _{n=1}^{x}1_{\Omega }(n)}{x}}\right),}$$ where ⁠${\displaystyle \Omega }$⁠ is a fixed list of prime numbers, ${\displaystyle 1_{\Omega }(n)=0}$ if at least one of the primes in ⁠${\displaystyle \Omega }$⁠ is a prime factor of ⁠${\displaystyle n}$⁠, and ${\displaystyle 1_{\Omega }(n)=1}$ otherwise. In particular, ⁠${\displaystyle \gamma (\emptyset )=\gamma }$⁠.

Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10²⁴⁴⁶⁶³.^[47][48] If e^γ is a rational number, then its denominator must be greater than 10¹⁵⁰⁰⁰.^[3]

Euler's constant is conjectured not to be an algebraic period,^[3] but the values of its first 10⁹ decimal digits seem to indicate that it could be a normal number.^[49]

The simple continued fraction expansion of Euler's constant is given by:^[50]

${\displaystyle \gamma =0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{4+\dots }}}}}}}}}}}}}}}$

which has no apparent pattern. It is known to have at least 16,695,000,000 terms,^[50] and it has infinitely many terms if and only if γ is irrational.

Numerical evidence suggests that both Euler's constant γ as well as the constant e^γ are among the numbers for which the geometric mean of their simple continued fraction terms converges to Khinchin's constant. Similarly, when ${\displaystyle p_{n}/q_{n}}$ are the convergents of their respective continued fractions, the limit ${\displaystyle \lim _{n\to \infty }q_{n}^{1/n}}$ appears to converge to Lévy's constant in both cases.^[51] However neither of these limits has been proven.^[52]

There also exists a generalized continued fraction for Euler's constant.^[53]

A good simple approximation of γ is given by the reciprocal of the square root of 3 or about 0.57735:^[54]

${\displaystyle {\frac {1}{\sqrt {3}}}=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+\dots }}}}}}}}}}}}}}}$

with the difference being about 1 in 7,429.

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

$${\displaystyle -\gamma =\Gamma '(1)=\psi (1).}$$

This is equal to the limits:

$${\displaystyle {\begin{aligned}-\gamma &=\lim _{z\to 0}\left(\Gamma (z)-{\frac {1}{z}}\right)\\&=\lim _{z\to 0}\left(\psi (z)+{\frac {1}{z}}\right).\end{aligned}}}$$

Further limit results are:^[55]

$${\displaystyle {\begin{aligned}\lim _{z\to 0}{\frac {1}{z}}\left({\frac {1}{\Gamma (1+z)}}-{\frac {1}{\Gamma (1-z)}}\right)&=2\gamma \\\lim _{z\to 0}{\frac {1}{z}}\left({\frac {1}{\psi (1-z)}}-{\frac {1}{\psi (1+z)}}\right)&={\frac {\pi ^{2}}{3\gamma ^{2}}}.\end{aligned}}}$$

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

$${\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left({\frac {\Gamma \left({\frac {1}{n}}\right)\Gamma (n+1)\,n^{1+{\frac {1}{n}}}}{\Gamma \left(2+n+{\frac {1}{n}}\right)}}-{\frac {n^{2}}{n+1}}\right)\\&=\lim \limits _{m\to \infty }\sum _{k=1}^{m}{m \choose k}{\frac {(-1)^{k}}{k}}\log {\big (}\Gamma (k+1){\big )}.\end{aligned}}}$$

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

$${\displaystyle {\begin{aligned}\gamma &=\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{m}}\\&=\log {\frac {4}{\pi }}+\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{2^{m-1}m}}.\end{aligned}}}$$ The constant ${\displaystyle \gamma }$ can also be expressed in terms of the sum of the reciprocals of non-trivial zeros ${\displaystyle \rho }$ of the zeta function:^[56]

${\displaystyle \gamma =\log 4\pi +\sum _{\rho }{\frac {2}{\rho }}-2}$

Other series related to the zeta function include:

$${\displaystyle {\begin{aligned}\gamma &={\tfrac {3}{2}}-\log 2-\sum _{m=2}^{\infty }(-1)^{m}\,{\frac {m-1}{m}}{\big (}\zeta (m)-1{\big )}\\&=\lim _{n\to \infty }\left({\frac {2n-1}{2n}}-\log n+\sum _{k=2}^{n}\left({\frac {1}{k}}-{\frac {\zeta (1-k)}{n^{k}}}\right)\right)\\&=\lim _{n\to \infty }\left({\frac {2^{n}}{e^{2^{n}}}}\sum _{m=0}^{\infty }{\frac {2^{mn}}{(m+1)!}}\sum _{t=0}^{m}{\frac {1}{t+1}}-n\log 2+O\left({\frac {1}{2^{n}\,e^{2^{n}}}}\right)\right).\end{aligned}}}$$

The error term in the last equation is a rapidly decreasing function of n. 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:^[57]

$${\displaystyle {\begin{aligned}\gamma &=\lim _{s\to 1^{+}}\sum _{n=1}^{\infty }\left({\frac {1}{n^{s}}}-{\frac {1}{s^{n}}}\right)\\&=\lim _{s\to 1}\left(\zeta (s)-{\frac {1}{s-1}}\right)\\&=\lim _{s\to 0}{\frac {\zeta (1+s)+\zeta (1-s)}{2}}\end{aligned}}}$$

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

$${\displaystyle \gamma =\lim _{n\to \infty }{\frac {1}{n}}\,\sum _{k=1}^{n}\left(\left\lceil {\frac {n}{k}}\right\rceil -{\frac {n}{k}}\right)}$$

where ⌈ ⌉ are ceiling brackets. This formula indicates that when taking any positive integer n and dividing it by each positive integer k less than n, the average fraction by which the quotient n/k falls short of the next integer tends to γ (rather than 0.5) as n 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:

$${\displaystyle \gamma =\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n-\sum _{m=2}^{\infty }{\frac {\zeta (m,n+1)}{m}}\right),}$$

where ζ(s, k) is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, H_n. Expanding some of the terms in the Hurwitz zeta function gives:

$${\displaystyle H_{n}=\log(n)+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-\varepsilon ,}$$ where 0 < ε < ⁠1/252n⁶⁠.

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

$${\displaystyle \gamma =12\,\log(A)-\log(2\pi )+{\frac {6}{\pi ^{2}}}\,\zeta '(2)}$$

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

$${\displaystyle \gamma =\lim _{n\to \infty }\left(-n+\zeta \left({\frac {n+1}{n}}\right)\right)}$$

Numerous formulations have been derived that express ${\displaystyle \gamma }$ in terms of sums and logarithms of triangular numbers.^{[58][59][60][61]} One of the earliest of these is a formula^[62][63] for the ${\displaystyle n}$th harmonic number attributed to Srinivasa Ramanujan where ${\displaystyle \gamma }$ is related to ${\displaystyle \textstyle \ln 2T_{k}}$ in a series that considers the powers of ${\displaystyle \textstyle {\frac {1}{T_{k}}}}$ (an earlier, less-generalizable proof^[64][65] by Ernesto Cesàro gives the first two terms of the series, with an error term):

${\displaystyle {\begin{aligned}\gamma &=H_{u}-{\frac {1}{2}}\ln 2T_{u}-\sum _{k=1}^{v}{\frac {R(k)}{T_{u}^{k}}}-\Theta _{v}\,{\frac {R(v+1)}{T_{u}^{v+1}}}\end{aligned}}}$

From Stirling's approximation^[58][66] follows a similar series:

${\displaystyle \gamma =\ln 2\pi -\sum _{k=2}^{\infty }{\frac {\zeta (k)}{T_{k}}}}$

The series of inverse triangular numbers also features in the study of the Basel problem^[67][68] posed by Pietro Mengoli. Mengoli proved that ${\displaystyle \textstyle \sum _{k=1}^{\infty }{\frac {1}{2T_{k}}}=1}$, a result Jacob Bernoulli later used to estimate the value of ${\displaystyle \zeta (2)}$, placing it between ${\displaystyle 1}$ and ${\displaystyle \textstyle \sum _{k=1}^{\infty }{\frac {2}{2T_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{T_{k}}}=2}$. This identity appears in a formula used by Bernhard Riemann to compute roots of the zeta function,^[69] where ${\displaystyle \gamma }$ is expressed in terms of the sum of roots ${\displaystyle \rho }$ plus the difference between Boya's expansion and the series of exact unit fractions ${\displaystyle \textstyle \sum _{k=1}^{\infty }{\frac {1}{T_{k}}}}$:

${\displaystyle \gamma -\ln 2=\ln 2\pi +\sum _{\rho }{\frac {2}{\rho }}-\sum _{k=1}^{\infty }{\frac {1}{T_{k}}}}$

γ equals the value of a number of definite integrals:

$${\displaystyle {\begin{aligned}\gamma &=-\int _{0}^{\infty }e^{-x}\log x\,dx\\&=-\int _{0}^{1}\log \left(\log {\frac {1}{x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{x\cdot e^{x}}}\right)dx\\&=\int _{0}^{1}{\frac {1-e^{-x}}{x}}\,dx-\int _{1}^{\infty }{\frac {e^{-x}}{x}}\,dx\\&=\int _{0}^{1}\left({\frac {1}{\log x}}+{\frac {1}{1-x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{1+x^{k}}}-e^{-x}\right){\frac {dx}{x}},\quad k>0\\&=2\int _{0}^{\infty }{\frac {e^{-x^{2}}-e^{-x}}{x}}\,dx,\\&=\log {\frac {\pi }{4}}-\int _{0}^{\infty }{\frac {\log x}{\cosh ^{2}x}}\,dx,\\&=\int _{0}^{1}H_{x}\,dx,\\&={\frac {1}{2}}+\int _{0}^{\infty }\log \left(1+{\frac {\log \left(1+{\frac {1}{t}}\right)^{2}}{4\pi ^{2}}}\right)dt\\&=1-\int _{0}^{1}\{1/x\}dx\\&={\frac {1}{2}}+\int _{0}^{\infty }{\frac {2x\,dx}{(x^{2}+1)(e^{2\pi x}-1)}}\end{aligned}}}$$ where H_x is the fractional harmonic number, and ${\displaystyle \{1/x\}}$ is the fractional part of ${\displaystyle 1/x}$.

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

$${\displaystyle {\begin{aligned}&\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{xe^{x}}}\right)dx=\int _{0}^{\infty }{\frac {e^{-x}+x-1}{x[e^{x}-1]}}dx=\int _{0}^{\infty }{\frac {1}{x[e^{x}-1]}}\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m+1}}{(m+1)!}}dx\\[2pt]&=\int _{0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![e^{x}-1]}}dx=\sum _{m=1}^{\infty }\int _{0}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![e^{x}-1]}}dx=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}\int _{0}^{\infty }{\frac {x^{m}}{e^{x}-1}}dx\\[2pt]&=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}m!\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\sum _{n=1}^{\infty }{\frac {1}{n^{m+1}}}=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}\\[2pt]&=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}=\sum _{n=1}^{\infty }\left[{\frac {1}{n}}-\log \left(1+{\frac {1}{n}}\right)\right]=\gamma \end{aligned}}}$$

The integral on the second line of the equation is the definition of the Riemann zeta function, which is m!ζ(m + 1).

Definite integrals in which γ appears include:^[2][13]

$${\displaystyle {\begin{aligned}\int _{0}^{\infty }e^{-x^{2}}\log x\,dx&=-{\frac {(\gamma +2\log 2){\sqrt {\pi }}}{4}}\\\int _{0}^{\infty }e^{-x}\log ^{2}x\,dx&=\gamma ^{2}+{\frac {\pi ^{2}}{6}}\\\int _{0}^{\infty }{\frac {e^{-x}\log x}{e^{x}+1}}\,dx&={\frac {1}{2}}\log ^{2}2-\gamma \end{aligned}}}$$

We also have Catalan's 1875 integral^[70]

$${\displaystyle \gamma =\int _{0}^{1}\left({\frac {1}{1+x}}\sum _{n=1}^{\infty }x^{2^{n}-1}\right)\,dx.}$$

One can express γ using a special case of Hadjicostas's formula as a double integral^[40][71] with equivalent series:

$${\displaystyle {\begin{aligned}\gamma &=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-xy)\log xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\log {\frac {n+1}{n}}\right).\end{aligned}}}$$

An interesting comparison by Sondow^[71] is the double integral and alternating series

$${\displaystyle {\begin{aligned}\log {\frac {4}{\pi }}&=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1+xy)\log xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left((-1)^{n-1}\left({\frac {1}{n}}-\log {\frac {n+1}{n}}\right)\right).\end{aligned}}}$$

It shows that log ⁠4/π⁠ may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series^[72]

$${\displaystyle {\begin{aligned}\gamma &=\sum _{n=1}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}\\\log {\frac {4}{\pi }}&=\sum _{n=1}^{\infty }{\frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}},\end{aligned}}}$$

where N₁(n) and N₀(n) are the number of 1s and 0s, respectively, in the base 2 expansion of n.

In general,

$${\displaystyle \gamma =\lim _{n\to \infty }\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\ldots +{\frac {1}{n}}-\log(n+\alpha )\right)\equiv \lim _{n\to \infty }\gamma _{n}(\alpha )}$$

for any α > −n. However, the rate of convergence of this expansion depends significantly on α. In particular, γ_n(1/2) exhibits much more rapid convergence than the conventional expansion γ_n(0).^[73][74] This is because

$${\displaystyle {\frac {1}{2(n+1)}}<\gamma _{n}(0)-\gamma <{\frac {1}{2n}},}$$

while

$${\displaystyle {\frac {1}{24(n+1)^{2}}}<\gamma _{n}(1/2)-\gamma <{\frac {1}{24n^{2}}}.}$$

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 γ: $${\displaystyle \gamma =\sum _{k=1}^{\infty }\left({\frac {1}{k}}-\log \left(1+{\frac {1}{k}}\right)\right).}$$

The series for γ is equivalent to a series Nielsen found in 1897:^[55