Harald Helfgott

Optimal bounds for sums of arithmetic functions

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 for 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 a 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. We also give a sharp result on such sums for a_n non-negative and not necessarily bounded, and apply it to obtain optimal bounds on psi(x)-x given finite verifications of RH.

Our proofs mixe a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type as in (Graham--Vaaler, 1981) and (Carneiro--Littmann, 2013). While our approach does not depend on existing explicit work in number theory, our method has an important step in common with work on another problem by (Ramana–Ramare, 2020).