EulerMascheroni

EulerMascheroni refers to the Euler–Mascheroni constant, usually denoted by gamma. It is defined as the limiting difference between the harmonic series and the natural logarithm: gamma = lim_{n→∞} (H_n − ln n), where H_n = sum_{k=1}^n 1/k. The constant is named after Leonhard Euler and Lorenzo Mascheroni, who studied the relationship between harmonic numbers and logarithms in the 18th century.

Beyond the limit definition, gamma has several equivalent representations. It can be expressed as γ = −Γ′(1), linking it to the derivative of the gamma function. It also appears in the Laurent series of the Riemann zeta function about s = 1: ζ(s) = 1/(s−1) + γ + ..., and it features in various asymptotic formulas such as those produced by the Euler–Maclaurin summation formula.

Numerically, gamma is approximately 0.577215664901532860606512090082... The constant is known to many digits, but it remains unknown whether it is irrational or transcendental.

Historically, the constant is associated with Euler and Mascheroni due to their early work on the behavior of harmonic series and logarithms. The name Euler–Mascheroni is used to honor both mathematicians for their contributions to the study of this constant.

Significance of the Euler–Mascheroni constant extends across analysis and number theory. It arises in the asymptotics of factorials via the gamma function, in integrals involving e^x/x, and in the study of analytic properties of zeta and related functions, making it a central constant in the theory of special functions and harmonic sums.