Optimal bounds for sums of non-negative arithmetic functions

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series admitting meromorphic continuation to the complex plane. Assume we know the location of the poles of $A(s)$ with $|\Im s| \leq T$, and their residues, for some large constant $T$. It is natural to ask how such finite spectral information may be best used to estimate partial sums $\sum_{n\leq x} a_n$. Here, we prove a sharp, general result on sums $\sum_{n\leq x} a_n n^{-\sigma}$ for $a_n$ non-negative, giving an optimal way to use information on the poles of $A(s)$ with $|\Im s|\leq T$, with no need for zero-free regions. We give not just bounds, but an explicit formula with compact support. Our bounds on $\psi(x)-x$ are, unsurprisingly, better and often simpler than a long list of existing explicit versions of the Prime Number Theorem. We treat the case of $M(x)$ and similar functions in a companion paper. Our solution mixes a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type found in (Graham--Vaaler, 1981).