Stieltjes constants

In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function: The constant is known as the Euler–Mascheroni constant. The Stieltjes constants are given by the limit (In the case n = 0, the first summand requires evaluation of 00, which is taken to be 1.) Cauchy's differentiation formula leads to the integral representation Various representations in terms of integrals and infinite series are given in works of Jensen, Franel, Hermite, Hardy, Ramanujan, Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some other authors. In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that where δn,k is the Kronecker symbol (Kronecker delta). Among other formulae, we find see. As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912 Israilov gave semi-convergent series in terms of Bernoulli numbers Connon, Blagouchine and Coppo gave several series with the binomial coefficients where Gn are Gregory's coefficients, also known as reciprocal logarithmic numbers (G1=+1/2, G2=−1/12, G3=+1/24, G4=−19/720,... ). More general series of the same nature include these examples and or where ψn(a) are the Bernoulli polynomials of the second kind and Nn,r(a) are the polynomials given by the generating equation respectively (note that Nn,1(a) = ψn(a)). Oloa and Tauraso showed that series with harmonic numbers may lead to Stieltjes constants Blagouchine obtained slowly-convergent series involving unsigned Stirling numbers of the first kind as well as semi-convergent series with rational terms only where m=0,1,2,... In particular, series for the first Stieltjes constant has a surprisingly simple form where Hn is the nth harmonic number. More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, Coffey. The Stieltjes constants satisfy the bound given by Berndt in 1972.