Математика Этерна
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On connection between values of Riemann zeta function at rationals and generalized harmonic numbers

Pawe˜J. Szab˜owski

Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer and rational values of arguments to certain rapidly converging series, where some generalized harmonic numbers appear. Most of the results of the paper can be derived from the recent, more advanced results, on the properties of Arakawa-Kaneko zeta functions. We derive our results directly, by solving simple recursions. The form of mentioned above generalized harmonic numbers carries information, about the values of the arguments of Hurwitz function. In particular we prove: 8k 2 N : (k; 1) = (k) = 2 k1 2 k11 P1 n=1 H (k1) n n2n ; where H (k) n are deÖned below generalized harmonic numbers, or that K = P1 n=0 n!(H2n+1Hn=2) 2(2n+1)!! ; where K denotes Calatan constant and Hn denotes nth (ordinary) harmonic number. Further we show that generating function of the numbers ^(k) = P1 j=1(1)j1=jk , k 2 N and ^(0) = 1=2 is equal to B(1=2; 1 y; 1 + y) where B(x; a; b) denotes incomplete beta