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We exploit some properties of the Hurwitz zeta function $\zeta (n,x)$ inorder 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 algebraicnumbers. 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 orderrespectively $2m+1$ and $2m+2$ such that their $n-$th coefficients of sine andcosine Fourier transforms are equal to $% (-1)^{n}/n^{2m+1}$ and$(-1)^{n}/n^{2m+2}$ respectively.

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