Large central values of Dirichlet L-functions in cosets
Ivan Ermoshin
TL;DR
The paper analyzes extreme central values $L(\chi,1/2)$ of Dirichlet $L$-functions within cosets of the character group modulo $q$ by extending Soundararajan's resonator method to unitary families. It derives lower bounds for the maximal central value in both trivial and non-trivial cosets, with the main exponent scaling like $\mathcal L=\sqrt{\log q/\log\log q}$ and corrections depending on the coset index $[\mathcal X:H]$ and parity structure. The approach combines an optimized resonator (via $M_1,M_2$ and the approximate functional equation) with a careful orthogonality analysis and a congruence-nonexistence estimate to control off-diagonal terms, yielding explicit index-dependent extreme-value bounds. The results illuminate how coset structure influences the distribution of large central $L$-values and point toward potential removal of index restrictions with further refinement.
Abstract
In this paper we consider the distribution of large central values of Dirichlet L-functions over cosets of the group of characters modulo q via Soundararajan's resonator method.