Optimal bounds for sums of non-negative arithmetic functions
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arXiv
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| Format: | Preprint |
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2025
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| author | Chirre, Andrés Helfgott, Harald Andrés |
| author_facet | Chirre, Andrés Helfgott, Harald Andrés |
| contents | 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 $\sum_{n\leq x} a_n n^{-σ}$ 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 $ψ(x)-x$ 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15709 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal bounds for sums of non-negative arithmetic functions Chirre, Andrés Helfgott, Harald Andrés Number Theory 11N37, 11M26, 42A38, 42A05 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 $\sum_{n\leq x} a_n n^{-σ}$ 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 $ψ(x)-x$ 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). |
| title | Optimal bounds for sums of non-negative arithmetic functions |
| topic | Number Theory 11N37, 11M26, 42A38, 42A05 |
| url | https://arxiv.org/abs/2512.15709 |