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:
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 (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. 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 appears, among other places, in the following (where '' means that this entry contains an explicit equation):
Expressions involving the exponential integral
The Laplace transform* of the natural logarithm
The first term of the Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants*
Calculations of the digamma function
A product formula for the gamma function
The asymptotic expansion of the gamma function for small arguments.
An inequality for Euler's totient function
The growth rate of the divisor function
In dimensional regularization of Feynman diagrams in quantum field theory
The calculation of the Meissel–Mertens constant
The third of Mertens' theorems*
Solution of the second kind to Bessel's equation
In the regularization/renormalization of the harmonic series as a finite value
The mean of the Gumbel distribution
The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named.
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: Starting from n = 1, the sequence of harmonic numbers begins: Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory.
Cyanobacterial blooms occur at increasing frequency and intensity, notably in freshwater. This leads to the introduction of complex mixtures of their products, i.e., cyano-metabolites, to drinking water treatment plants. To assess the fate of cyano-metabol ...
Amer Chemical Soc2024
Related lectures (36)
We prove a fractional Hardy-Rellich inequality with an explicit constant in bounded domains of class C-1,C-1. The strategy of the proof generalizes an approach pioneered by E. Mitidieri (Mat. Zametki, 2000) by relying on a Pohozaev-type identity. ...
Pergamon-Elsevier Science Ltd2024
We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the polynomial ring g ...