A heuristic derivation of linear recurrence relations for $\zeta '(-2k)$ and $\zeta(2k+1)$

TL;DR

This paper derives linear recurrence relations for derivatives of the Riemann zeta function at negative even integers and for zeta at positive odd integers using historical methods involving harmonic sums, without relying on modern analytic continuation.

Contribution

It introduces a heuristic approach to obtain recurrence relations for $ ext{zeta}'(-2k)$ and $ ext{zeta}(2k+1)$ based on formal manipulations of harmonic sums, revisiting classical methods.

Findings

Derived recurrence relations for $ ext{zeta}'(-2k)$ and $ ext{zeta}(2k+1)$.

Connected harmonic sum manipulations to zeta function derivatives.

Provided formulas that could facilitate computation of zeta values.

Abstract

We have gone back to old methods found in the historical part of Hardy's Divergent Series well before the invention of the modern analytic continuation to use formal manipulation of harmonic sums which produce some interesting formulae. These are linear recurrence relations for $\displaystyle{ \sum_{n=1}^\infty H_n n^k}$ which in turn yield linear recurrence relations for $\zeta '(-k)$ and hence using the functional equation to a linear recurrence relation for $\zeta '(2k)$ and $\zeta (2k+1)$. Questions of rigor have been postponed to a subsequent preprint.