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We have gone back to old methods found in the historical part of Hardy'sDivergent Series well before the invention of the modern analytic continuationto use formal manipulation of harmonic sums which produce some interestingformulae. 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 recurrencerelation for $\zeta '(2k)$ and $\zeta (2k+1)$. Questions of rigor have beenpostponed to a subsequent preprint.

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