Optimal bounds for sums of bounded arithmetic functions

TL;DR

This paper develops optimal bounds for sums of bounded arithmetic functions using spectral data from Dirichlet series, significantly improving bounds on the Mertens function and providing explicit formulas with pole contributions.

Contribution

It introduces a sharp, general method to estimate sums of bounded arithmetic functions from limited spectral data, improving bounds on M(x) and clarifying the contribution of each pole.

Findings

Stronger bounds on M(x) than previous results.

Explicit formula for M(x) with clear pole contributions.

Method combines Fourier analysis, contour-shifting, and Beurling--Selberg approximants.

Abstract

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of $A(s)$ with $|\Im s| \leq T$ for some large constant $T$. What is the best way to use such finite spectral data to give explicit estimates on sums $\sum_{n\leq x} a_n$? The problem of giving explicit bounds on the Mertens function $M(x) = \sum_{n\leq x} \mu(n)$ illustrates how open this basic question was. Bounding $M(x)$ might seem equivalent to estimating $\psi(x) = \sum_{n\leq x} \Lambda(n)$ or the number of primes $\leq x$. 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 $\sum_{n\leq x} a_n n^{-\sigma}$ for $a_n$ bounded, giving an optimal way to use information on the poles of $A(s)$ with $|\Im s|\leq T$ 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 $\psi(x)$ are given in a companion paper.) Using rigorous residue computations by D. Platt, we obtain, for $x\geq 1$, $$|M(x)|\leq \frac{3}{\pi\cdot 10^{10}}\cdot x + 11.39 \sqrt{x}.$$ 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 $\sigma=1$, the approximant in (Vaaler, 1985) reappears. While we proceed independently of existing explicit work on $M(x)$ and $\psi(x)$, our method has an important step in common with work on another problem by (Ramana--Ramar\'e, 2020).