Prime Number Error Terms

TL;DR

This paper explores conjectures on the size of prime number error terms, generalizes existing conjectures, and adapts methods to establish lower bounds, also providing an $L^2$ bound for almost periodic functions.

Contribution

It introduces a general conjecture encompassing Montgomery's and M"{o}bius function error terms, and adapts Lamzouri's approach to establish lower bounds under a generalized linear independence hypothesis.

Findings

A general conjecture for prime number error terms is proposed.

A lower bound for these error terms is established under a generalized linear independence conjecture.

An $L^2$ bound for almost periodic functions is proved, improving existing results.

Abstract

In 1980 Montgomery made a conjecture about the true order of the error term in the prime number theorem. In 2012 the author made an analogous conjecture for the true order of the sum of the M\"{o}bius function, $M(x)$. This refined an earlier conjecture of Gonek from the 1990's. In this article we speculate on the true size of a large class of prime number error terms and present a general conjecture. This general conjecture includes both Montgomery's conjecture and the conjecture for $M(x)$ as special cases. Recently, Lamzouri (Springer volume: Essays in Analytic Number Theory, In Honor of Helmut Maier's 70th birthday) showed that an effective linear independence conjecture (ELI) for the zeros of the zeta function implies one of the inequalities in Montgomery's conjecture. In this article we adapt Lamzouri's method to show that a generalized effective linear independence (GELI) conjecture implies a lower bound for general prime number error terms. Furthermore, of independent interest, we prove an $L^2$ bound for almost periodic functions. This allows us to weaken significantly one of the conditions in Lamzouri's main result and also give an improvement of the main theorem in an article of Akbary-Ng-Shahabi (Q. J. Math. 65 (2014), no. 3).