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:
γ = lim n → ∞ ( ∑ k = 1 n 1 k − log n ) = ∫ 1 ∞ ( 1 ⌊ x ⌋ − 1 x ) d x . {\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n\right)\\&=\int _{1}^{\infty }\left({\frac {1}{\lfloor x\rfloor }}-{\frac {1}{x}}\right)\,\mathrm {d} x.\end{aligned}}}
Here, ⌊·⌋ represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is:
Introduction The usual definition of Euler's constant is, as stated above, given via the limiting difference between the harmonic series and the natural logarithm. The divergence of the former series was already known centuries prior, and Euler, who sought to find a continuous interpolation of the discrete harmonic series, found its growth rate to be asymptotically equivalent to that of the natural logarithm. In particular, he proved that:
γ = lim n → ∞ ( 1 1 + 1 2 + 1 3 + … + 1 n − log ( n ) ) {\displaystyle \gamma =\lim _{n\to \infty }\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\ldots +{\frac {1}{n}}-\log(n)\right)}
exists. This can be seen by setting a n = H n − log n {\displaystyle a_{n}=H_{n}-\log n} as the partial terms of the limit (Hn are the harmonic numbers) and noting that this sequence is monotonically decreasing:
a n + 1 − a n = 1 n + 1 − log ( 1 + 1 n ) < 0 {\displaystyle a_{n+1}-a_{n}={\frac {1}{n+1}}-\log \left(1+{\frac {1}{n}}\right)<0}
as well as bounded from below:
a n = ∑ k = 1 n 1 k − ∫ 1 n 1 t d t > 0. {\displaystyle a_{n}=\sum _{k=1}^{n}{\frac {1}{k}}-\int _{1}^{n}{\frac {1}{t}}\,dt>0.}
and hence convergent to a constant 0 ≤ γ < 1 = a 1 {\displaystyle 0\leq \gamma <1=a_{1}} . Euler showed that the following infinite series approaches γ:
γ = ∑ k = 1 ∞ ( 1 k − log ( 1 + 1 k ) ) . {\displaystyle \gamma =\sum _{k=1}^{\infty }\left({\frac {1}{k}}-\log \left(1+{\frac {1}{k}}\right)\right).}
This can be seen by noting that its partial sums form a telescoping series, such that the series again yields a limit between the harmonic series and a shifted natural logarithm. In general:
γ = lim n → ∞ ( 1 1 + 1 2 + 1 3 + … + 1 n − log ( n + α ) ) ≡ lim n → ∞ γ n ( α ) {\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, γn(1/2) exhibits much more rapid convergence than the conventional expansion γn(0). This is because
1 2 ( n + 1 ) < γ n ( 0 ) − γ < 1 2 n , {\displaystyle {\frac {1}{2(n+1)}}<\gamma _{n}(0)-\gamma <{\frac {1}{2n}},}
while
1 24 ( n + 1 ) 2 < γ n ( 1 / 2 ) − γ < 1 24 n 2 . {\displaystyle {\frac {1}{24(n+1)^{2}}}<\gamma _{n}(1/2)-\gamma <{\frac {1}{24n^{2}}}.}
This can also be seen in the following asymptotic formulas:
γ ∼ H n − log n − 1 2 n + 1 12 n 2 − 1 120 n 4 + ⋯ {\textstyle \gamma \sim H_{n}-\log n-{\frac {1}{2n}}+{\frac {1}{12n^{2}}}-{\frac {1}{120n^{4}}}+\cdots } (Euler)
γ ∼ H n − log ( n + 1 2 + 1 24 n − 1 48 n 2 + ⋯ ) {\textstyle \gamma \sim H_{n}-\log \left({n+{\frac {1}{2}}+{\frac {1}{24n}}-{\frac {1}{48n^{2}}}+\cdots }\right)} (Negoi)
γ ∼ H n − log n + log ( n + 1 ) 2 − 1 6 n ( n + 1 ) + 1 30 n 2 ( n + 1 ) 2 − ⋯ {\textstyle \gamma \sim H_{n}-{\frac {\log n+\log(n+1)}{2}}-{\frac {1}{6n(n+1)}}+{\frac {1}{30n^{2}(n+1)^{2}}}-\cdots } (Cesàro) The third formula is also called the Ramanujan expansion. Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.
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". 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. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835, and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842. Other notations were also occasionally used. 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 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.
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, Γ ( 1 / x ) ∼ x − γ {\displaystyle \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 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 makes the irrationality of γ 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 γ and the Gompertz constant δ is irrational; Tanguy Rivoal proved in 2012 that at least one of them is transcendental. Kurt Mahler showed in 1968 that the number π 2 Y 0 ( 2 ) J 0 ( 2 ) − γ {\textstyle {\frac {\pi }{2}}{\frac {Y_{0}(2)}{J_{0}(2)}}-\gamma } is transcendental, where J 0 {\displaystyle J_{0}} and Y 0 {\displaystyle Y_{0}} are the usual Bessel functions. It is known that the transcendence degree of the field Q ( e , γ , δ ) {\displaystyle \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
γ ( a , q ) = lim n → ∞ ( ∑ k = 0 n 1 a + k q − log ( a + n q ) q ) {\displaystyle \gamma (a,q)=\lim _{n\rightarrow \infty }\left(\sum _{k=0}^{n}{\frac {1}{a+kq}}-{\frac {\log {(a+nq})}{q}}\right)}
is algebraic, if q ≥ 2 and 1 ≤ a < q; this family includes the special case γ(2,4) = γ/4. 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
γ ( Ω ) = lim x → ∞ ( ∑ n = 1 x 1 Ω ( n ) n − log x ⋅ lim x → ∞ ∑ n = 1 x 1 Ω ( n ) x ) , {\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 Ω {\displaystyle \Omega } is a fixed list of prime numbers, 1 Ω ( n ) = 0 {\displaystyle 1_{\Omega }(n)=0} if at least one of the primes in Ω {\displaystyle \Omega } is a prime factor of n {\displaystyle n} , and 1 Ω ( n ) = 1 {\displaystyle 1_{\Omega }(n)=1} otherwise. In particular, γ ( ∅ ) = γ {\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. If eγ 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:
γ = 0 + 1 1 + 1 1 + 1 2 + 1 1 + 1 2 + 1 1 + 1 4 + … {\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, 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 eγ 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 {\displaystyle p_{n}/q_{n}} are the convergents of their respective continued fractions, the limit lim n → ∞ q n 1 / n {\displaystyle \lim _{n\to \infty }q_{n}^{1/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 γ is given by the reciprocal of the square root of 3 or about 0.57735:
1 3 = 0 + 1 1 + 1 1 + 1 2 + 1 1 + 1 2 + 1 1 + 1 2 + …