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

Abstract

We exploit some properties of the Hurwitz zeta function $\zeta (n,x)$ in order to study sums of the form $\frac{1}{{\pi }^n}{\sum }_{j=-\infty }^{\infty }1/(jk+l{)}^n$ and $\frac{1}{{\pi }^n}{\sum }_{j=-\infty }^{\infty }(-1{)}^j/(jk+l{)}^n$ for and integer $l\le k/2$. We show that these sums are algebraic numbers. We also show that $1<n\in \mathbb{N}$ and $p\in \mathbb{Q}\cap (0,1)$ $:$ the numbers $(\zeta (n,p)+(-1{)}^n\zeta (n,1-p))/{\pi }^n$ are algebraic. On the way we find polynomials $s_m$ and $c_m$ of order respectively $2m+1$ and $2m+2$ such that their $n-$th coefficients of sine and cosine Fourier transforms are equal to and $(-1{)}^n/n^{2m+2}$ respectively.

16 pages