Mathematica Eterna
Open Access

Abstract

A few remarks on values of Hurwitz Zeta function at natural and rational arguments

Pawe˜J. Szab˜owski

We exploit some properties of the Hurwitz zeta function (n; x) in order to study sums of the form 1 n P1 j=1 1=(jk + l) n and 1 n P1 j=1(1)j=(jk + l) n for 2  n; k 2 N; and integer l  k=2. We show that these sums are algebraic numbers. We also show that 1 < n 2 N and p 2 Q \ (0; 1) : the numbers ((n; p) + (1)n (n; 1 p))=n are algebraic. On the way we Önd polynomials sm and cm of order respectively 2m + 1 and 2m + 2 such that their nth coe¢ cients of sine and cosine Fourier transforms are equal to (1)n=n2m+1 and (1)n=n2m+2 respe