2025/11/18 by Chirre, Andrés, Helfgott, Harald Andrés
Mathematics · #11M26 #11N37 #11Y35 #42A38 #Analytic Number Theory Research #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2511.14736
openalex publication_date 2025/11/18 · openalex created_date 2025/11/20 · openalex updated_date 2026/07/28
Let A(s) = ∑n an n-s be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of A(s) with |\Im s| ≤ T for some large constant T. What is the best way to use such finite spectral data to give explicit estimates on sums ∑n≤ x an? The problem of giving explicit bounds on the Mertens function M(x) = ∑n≤ x μ(n) illustrates how open this basic question was. Bounding M(x) might seem equivalent to estimating ψ(x) = ∑n≤ x Λ(n) or the number of primes ≤ x. However, we have long had fairly good explicit bounds on prime counts, while bounding M(x) remained a notoriously stubborn problem. We prove a sharp, general result on sums ∑n≤ x an n-σ for an bounded, giving an optimal way to use information on the poles of A(s) with |\Im s|≤ T and no data on the poles above. Our bounds on M(x) are stronger than previous ones by many orders of magnitude. (Similar results for ψ(x) are given in a companion paper.) Using rigorous residue computations by D. Platt, we obtain, for x≥ 1, |M(x)|≤ \frac3π⋅ 1010⋅ x + 11.39 √(x). This is a corollary of our main result, essentially an explicit formula with the contribution of each pole clearly stated; we shall discuss how this finer structure can be useful. Our proof mixes a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type (Carneiro--Littmann, 2013); for σ=1, the approximant in (Vaaler, 1985) reappears. While we proceed independently of existing explicit work on M(x) and ψ(x), our method has an important step in common with work on another problem by (Ramana--Ramaré, 2020).