A Didactic Coefficientwise Prime–Zero Dictionary for log ζ

For each fixed $\sigma>1$ we expand $\log \zeta(\sigma+z)$ in powers of $z$ in two different ways: first through the Euler product, and second through the completed function $\xi$ and its zero factorization. Coefficient comparison yields, for every $n \geq 1$, an exact identity between a weighted prime-power sum and a shifted power sum over the nontrivial zeros of $\zeta$, together with an explicit archimedean correction term. In this form the identity gives a direct local dictionary between the prime presentation and the zero presentation of the same zeta object. We also record a simple truncation algorithm for approximating the first few shifted zero sums from finitely many prime powers. Although the identities themselves are assembled from classical ingredients, the present coefficientwise packaging appears to offer a particularly direct local comparison between the Euler and zero presentations of $\zeta$.