Lower Bounds on High Moments of Twisted Fourier coefficients of Modular Forms
Peng Gao, Liangyi Zhao
TL;DR
The paper establishes a sharp lower bound for the $2k$-th moment of twisted sums $\sum_{n\le x}\chi(n)\lambda(n)$ when averaged over primitive Dirichlet characters modulo a large prime, for $x\le q^{1/2}$ and $k\ge 2$, with the lower bound matching under GRH up to constants. The approach blends mean-value techniques for random multiplicative models with careful Euler-product analyses tied to cusp form $L$-functions, including the symmetric square $L$-function, and a dyadic prime-block decomposition to capture the main terms. The main contributions include unconditional lower bounds $S_k(q,x;f) \gg \varphi(q)x^k(\log q)^{(k-1)^2}$ and, under GRH, sharp asymptotics $S_k(q,x;f) \asymp \varphi(q)x^k(\log q)^{(k-1)^2}$ for the stated range, reflecting improvements over square-root cancellation in high moments. The work leverages a combination of random-model moment estimates, prime-sum asymptotics, and sieve-type arguments to control both main and tail contributions, with implications for low-lying twisted moments of modular forms.
Abstract
For any large prime $q$, $x \leq 1$ and any real $k\geq 2$, we prove a lower bound for the following $2k$-th moment \begin{equation*} \sum_{\substack{χ\in X_q^*}} \Big| \sum_{n\leq x} χ(n)λ(n)\Big|^{2k}, \end{equation*} where $X_q^*$ denotes the set of primitive Dirichlet characters modulo $q$ and $λ(n)$ the Fourier coefficients of a fixed modular form. The bound we obtain is sharp up to a constant factor under the generalized Riemann Hypothesis.