# A new proof of Hal\'asz's Theorem, and its consequences

## TL;DR

This paper presents a new, more flexible proof of Halász's Theorem, leading to simpler proofs of prime distribution results like Hoheisel's and Linnik's theorems, with potential for broader applications.

## Contribution

A novel proof of Halász's Theorem that enhances flexibility and applicability, enabling easier derivation of prime distribution results in short intervals and arithmetic progressions.

## Key findings

- New proof of Halász's Theorem with improved flexibility
- Simplified proofs of Hoheisel's and Linnik's Theorems
- Potential for extending asymptotic analysis to short intervals and progressions

## Abstract

Hal\'asz's Theorem gives an upper bound for the mean value of a multiplicative function $f$. The bound is sharp for general such $f$, and, in particular, it implies that a multiplicative function with $|f(n)|\le 1$ has either mean value $0$, or is "close to" $n^{it}$ for some fixed $t$. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to short intervals and to arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel's Theorem), and that there are always primes near to the start of an arithmetic progression (Linnik's Theorem).