Cancellation in sums over special sequences on $\mathbf{\rm{GL}_{m}}$ and their applications
Qiang Ma, Rui Zhang
TL;DR
This work analyzes cancellation in the Dirichlet coefficients $\lambda_{\pi}(n)$ of automorphic L-functions over additive Waring--Goldbach sequences on $\mathrm{GL}_m$, establishing nontrivial bounds for additive twists over primes that do not rely on the generalized Ramanujan conjecture or the nonexistence of Landau--Siegel zeros. The authors develop exponential-sum bounds for $\sum_{p\le x} \lambda_{\pi}(p) e(p^{k}\alpha) \log p$ and apply a Hardy--Littlewood circle method to bound sums over $\mathcal{J}_{1,3}(N)$ and $\mathcal{J}_{2,5}(N)$, obtaining bounds like $r_{\pi}(N) \ll_{\pi,B} N^{2}(\log N)^{-B}$ and $w_{\pi}(N) \ll_{\pi,B} N^{3/2}(\log N)^{-B}$; under Hypothesis S, sharper bounds follow and connections to adjoint and Rankin--Selberg $L$-functions are exploited. The paper also derives an analogue of Sato--Tate equidistribution in this additive setting and discusses an average form of the Goldbach conjecture, highlighting potential implications for higher-rank automorphic forms. Overall, the work blends circle-method techniques with automorphic L-function theory to extend cancellation phenomena to higher ranks and couple additive problems with deep multiplicative structures.
Abstract
Let $a(n)$ be the $n$-th Dirichlet coefficient of the automorphic $L$-function or the Rankin--Selberg $L$-function. We investigate the cancellation of $a(n)$ over sequences linked to the Waring--Goldbach problem, by establishing a nontrivial bound for the additive twisted sums over primes on ${\mathrm{GL}}_m .$ The bound does not depend on the generalized Ramanujan conjecture or the nonexistence of Landau--Siegel zeros. Furthermore, we present an application associated with the Sato--Tate conjecture and propose a conjecture about the Goldbach conjecture on average bound.