Series involving ζ(n)

Abstract

Recall that for integers n ≥ 2, ζ (n) is defined byOf course, ζ (1) is not defined, since is divergent. A well-known particular value is ζ (2) = π2/6: numerous alternative proofs of this fact have been presented in the Gazette, e.g. the recent notes [1], [2].

On the product

“…If the relation (for example see [3]) then implies (4). Here the lower estimate follows from and the upper estimate from .…”

“…If , the formula (10) in [1, p. 46] yields the last expression above. The other series relating to are also given in [3]. The series in (2) and diverge at , i.e.…”

“…Here the lower estimate follows from and the upper estimate from . If the relation (for example see [3]) then implies (4).…”