The typical size of character and zeta sums is $o(\sqrt{x})$

TL;DR

This paper establishes conjecturally sharp upper bounds showing that character and zeta sums typically exhibit better than square root cancellation, advancing understanding of their size and moments in number theory.

Contribution

It proves new conjecturally sharp upper bounds for character and zeta sums, demonstrating better than squareroot cancellation phenomena and introducing novel conditioning techniques.

Findings

Character sums are typically o(√x) in size.

Zeta sums and related functions also show better than squareroot bounds.

New conditioning methods simplify the analysis of moments.

Abstract

We prove conjecturally sharp upper bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real. In particular, if both $x$ and $r/x$ tend to infinity with $r$ then $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)| = o(\sqrt{x})$, and so the sums $\sum_{n \leq x} \chi(n)$ typically exhibit "better than squareroot cancellation". We prove analogous better than squareroot bounds for the moments $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{it}|^{2q} dt$ of zeta sums; of Dirichlet theta functions $\theta(1,\chi)$; and of the sums $\sum_{n \leq x} h(n) \chi(n)$, where $h(n)$ is any suitably bounded multiplicative function (for example the M\"{o}bius function $\mu(n)$). The proofs depend on similar better than squareroot cancellation phenomena for low moments of random multiplicative functions. An important ingredient is a reorganisation of the conditioning arguments from the random case, so that one only needs to "condition" on a small collection of fairly short prime number sums. The conditioned quantities arising can then be well approximated by twisted second moments, whose behaviour is the same for character and zeta sums as in the random case.