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

TL;DR

This paper investigates the algebraic nature of certain sums involving the Hurwitz zeta function at rational and natural arguments, revealing their algebraic properties and deriving related Fourier coefficient polynomials.

Contribution

It demonstrates that specific sums of Hurwitz zeta values at rational points are algebraic and constructs polynomials with particular Fourier transform coefficients.

Findings

Sums of Hurwitz zeta at rational points are algebraic.

Derived polynomials with specific Fourier coefficients.

Established algebraic nature of zeta-related sums.

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 $% 2\leq n,k\in \mathbb{N},$ and integer $l\leq k/2$. We show that these sums are algebraic numbers. We also show that $1<n\in \mathbb{N}$ and $p\in \mathbb{Q\cap (}0,1\mathbb{)}$ $:$ 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 $% (-1)^{n}/n^{2m+1}$ and $(-1)^{n}/n^{2m+2}$ respectively.