Lower bounds for low moments of character sums, I: Short sums with general multiplicative weights

Abstract

We establish sharp lower bounds for the Dirichlet character moments $\frac{1}{r-1}{\sum }_{\chi \text{mod}r}|{\sum }_{n\le x}\chi (n){|}^{2q}$, where $r$ is a large prime, $1\le x\le r^{0.499}$, and $0\le q\le 1$ is real. These match the better than squareroot cancellation upper bounds obtained in previous work of the author. We prove the same sharp lower bounds for the moments $\frac{1}{T}{\int }_0^T|{\sum }_{n\le x}{n}^{it}{|}^{2q}dt$ of zeta sums, and more generally for moments of character sums ${\sum }_{n\le x}h(n)\chi (n)$ with suitably bounded multiplicative twist $h(n)$. The proofs are based on a comparison of the sizes of $\frac{1}{r-1}{\sum }_{\chi \text{mod}r}({\sum }_{n\le x}\chi (n))\overline{I(\chi )}$, $\frac{1}{r-1}{\sum }_{\chi \text{mod}r}|I(\chi ){|}^2$ and $\frac{1}{r-1}{\sum }_{\chi \text{mod}r}|I(\chi ){|}^4$, where $I(\chi )$ is a certain ``barrier adjusted'' Perron integral inspired by the analogous results for random multiplicative functions. In a companion paper, we extend these arguments to the full interesting range $x\le 0.99r$ for the unweighted character sum moments $\frac{1}{r-1}{\sum }_{\chi \text{mod}r}|{\sum }_{n\le x}\chi (n){|}^{2q}$. This leads to a positive proportion non-vanishing result for Dirichlet theta functions $\theta (1,\chi )$.

45 pages, including 15 page introduction