Pointwise and correlation bounds on Dedekind sums over small subgroups (Q6554729)

Dedekind sums are certain sums of products of sawtooth functions, which play an important role in analytic number theory, in particular in the context of modular forms. In this paper, the authors prove (point-wise as well as averaged) bounds on Dedekind sums under certain assumptions on the parameters (more specifically, the bounds apply to Dedekind sums \(s(\lambda,p)\), where \(\lambda\) is assumed to have small multiplicative order modulo \(p\)). The authors show that their results are essentially optimal, under the assumption that there exist infinitely many Mersenne primes. They also show how to relate their results on Dedekind sums to problems concerning high moments of Dirichlet \(L\)-functions \(L(1,\chi)\) over subgroups of characters. Two other recent papers dealing with closely related topics are [\textit{S. R. Louboutin} and \textit{M. Munsch}, Can. J. Math. 75, No. 5, 1711--1743 (2023; Zbl 1540.11036); \textit{M. Munsch} and \textit{I. E. Shparlinski}, ``Moments and non-vanishing of $L$-functions over thin subgroups'', Preprint, \url{arXiv:2309.10207}].