2025/12/17 by Chirre, Andrés, Helfgott, Harald Andrés
Mathematics · #11M26 #11N37 #42A05 #42A38 #Analytic Number Theory Research #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2512.15709
openalex publication_date 2025/12/17 · openalex created_date 2025/12/19 · openalex updated_date 2026/07/28
Let A(s) = ∑n an 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| ≤ 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 ∑n≤ x an. Here, we prove a sharp, general result on sums ∑n≤ x an n-σ for an non-negative, giving an optimal way to use information on the poles of A(s) with |\Im s|≤ T, with no need for zero-free regions. We give not just bounds, but an explicit formula with compact support. Our bounds on ψ(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).