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:

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. 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.

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):

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 ${\textstyle {\frac {\pi }{2}}{\frac {Y_{0}(2)}{J_{0}(2)}}-\gamma }$ is transcendental, where ${\displaystyle J_{0}}$ and ${\displaystyle Y_{0}}$ are the usual Bessel functions. It is known that the transcendence degree of the field ${\displaystyle \mathbb {Q} (e,\gamma ,\delta )}$ 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 $${\displaystyle \gamma (a,q)=\lim _{n\rightarrow \infty }\left(-{\frac {\log {(a+nq})}{q}}+\sum _{k=0}^{n}{\frac {1}{a+kq}}\right)}$$ is algebraic, if q ≥ 2 and 1 ≤ a < q; this family includes the special case γ(2,4) = γ/4.

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 $${\displaystyle \gamma (\Omega )=\lim _{x\rightarrow \infty }\left(\sum _{n=1}^{x}{\frac {1_{\Omega }(n)}{n}}-\log x\cdot \lim _{x\rightarrow \infty }{\frac {\sum _{n=1}^{x}1_{\Omega }(n)}{x}}\right),}$$ 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, ⁠${\displaystyle \gamma (\emptyset )=\gamma }$⁠.