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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. [3] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835, [ 4 ] and Augustus De Morgan used it in a textbook published in parts ...
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]
The interpolating function is in fact closely related to the digamma function = (+) +, where ψ(x) is the digamma function, and γ is the Euler–Mascheroni constant. The integration process may be repeated to obtain H x , 2 = ∑ k = 1 ∞ ( − 1 ) k − 1 k ( x k ) H k . {\displaystyle H_{x,2}=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}{x ...
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 :
γ is Euler–Mascheroni constant; τ = x + iy with y > 0. = / (), with q = e 2π i τ is the Dedekind eta function. So the Eisenstein series has a pole at s = 1 of residue π, and the (first) Kronecker limit formula gives the constant term of the Laurent series at this pole.
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 /
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It is almost certain that Euler meant that the sum of the reciprocals of the primes less than n is asymptotic to log log n as n approaches infinity. It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by Franz Mertens in 1874. [3] Thus Euler obtained a correct result by questionable means.