On squares of odd zeta values and analogues of Eisenstein series

Abstract

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for ζ(2m+1). The formula for ζ2(2m+1) is then generalized in two different directions, one, by considering the generalized divisor function σz(n), and the other, by studying a more general analogue of the aforementioned Eisenstein series, consisting of one more parameter N. A number of important special cases are derived from the first generalization. For example, we obtain a series representation for ζ(1+ω)ζ(−1−ω), where ω is a non-trivial zero of ζ(z). We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of ζ(4k−1) and ζ(4k+1) for k∈N.