Explicit estimates for sums of arithmetic functions, or the optimal use of finite information on Dirichlet series

Abstract: Let $F(s) = \sum_n a_n n^{-s}$ be a Dirichlet series. Say we have an analytic continuation of $F(s)$, and information on the poles of $F(s)$ with $|\Im s|\leq T$ for some large constant $T$. What is the best way to use this information 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. One might think that bounding $M(x)$ is essentially equivalent to estimating $\psi(x) = \sum_{n\leq x} \Lambda(n)$ or the number of primes $\leq x$. However, we have long had fairly satisfactory explicit bounds on $\psi(x)-x$, whereas bounding $M(x)$ well was a notoriously recalcitrant problem.

We give an optimal way to use information on the poles of $F(s)$ with $|\Im s|\leq T$. In particular, we give bounds on the Mertens function much stronger than those in the literature, while also substantially improving on estimates on $\psi(x)$.

We use functions of "Beurling-Selberg" type -- namely, an optimal approximant due to Carneiro-Littmann and an optional majorant/minorant due to Graham-Vaaler. Our procedure has points of contact with Wiener-Ikehara and also with work of Ramana and Ramaré, but does not rely on results in the explicit analytic-number-theory literature.

(joint work with Andrés Chirre)

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)