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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:
The area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant. In mathematics , the Stieltjes constants are the numbers γ k {\displaystyle \gamma _{k}} that occur in the Laurent series expansion of the Riemann zeta function :
The standard Gumbel distribution is the case where = and = with cumulative distribution function = ()and probability density function = (+).In this case the mode is 0, the median is ( ()), the mean is (the Euler–Mascheroni constant), and the standard deviation is /
The harmonic number with = ⌊ ⌋ (red line) with its asymptotic limit + (blue line) where is the Euler–Mascheroni constant.. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] = + + + + = =.
The definition for the gamma function due to Weierstrass is also valid for all complex numbers except non-positive integers: = = (+) /, where is the Euler–Mascheroni constant. [1] This is the Hadamard product of 1 / Γ ( z ) {\displaystyle 1/\Gamma (z)} in a rewritten form.
Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers (Abramowitz and Stegun 6.3.16): [1]
Secondary measures and the theory around them may be used to derive traditional formulas of analysis concerning the Gamma function, the Riemann zeta function, and the Euler–Mascheroni constant. They have also allowed the clarification of various integrals and series, although this tends to be difficult a priori.
where d represents the divisor function, and γ represents the Euler-Mascheroni constant. In 1898, Charles Jean de la Vallée-Poussin proved that if a large number n is divided by all the primes up to n, then the average fraction by which the quotient falls short of the next whole number is γ: