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 $\frac{1}{r-1} \sum_{χ\; \text{mod} \; r} |\sum_{n \leq x} χ(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r^{0.499}$, and $0 \leq q \leq 1$ 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 $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{it}|^{2q} dt$ of zeta sums, and more generally for moments of character sums $\sum_{n \leq x} h(n) χ(n)$ with suitably bounded multiplicative twist $h(n)$. The proofs are based on a comparison of the sizes of $\frac{1}{r-1} \sum_{χ\; \text{mod} \; r} (\sum_{n \leq x} χ(n)) \overline{I(χ)}$, $\frac{1}{r-1} \sum_{χ\; \text{mod} \; r} |I(χ)|^2$ and $\frac{1}{r-1} \sum_{χ\; \text{mod} \; r} |I(χ)|^4$, where $I(χ)$ 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 $x \leq 0.99r$ for the unweighted character sum moments $\frac{1}{r-1} \sum_{χ\; \text{mod} \; r} |\sum_{n \leq x} χ(n)|^{2q}$. This leads to a positive proportion non-vanishing result for Dirichlet theta functions $θ(1,χ)$.