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

Original · EN

We exploit some properties of the Hurwitz zeta function ζ(n,x) in order to study sums of the form 1πⁿ∑ⱼ₌₋∞∞1/(jk+l)ⁿ and 1πⁿ∑ⱼ₌₋∞∞(-1)ʲ/(jk+l)ⁿ for % 2≤ n,k∈ N, and integer l≤ k/2. We show that these sums are algebraic numbers. We also show that 1<n∈ N and p∈ Q∩ (0,1): the numbers (ζ(n,p)+(-1)ⁿζ(n,1-p))/πⁿ are algebraic. On the way we find polynomials sₘ and cₘ 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/n²ᵐ⁺¹ and (-1)ⁿ/n²ᵐ⁺² respectively.