Mean square of inverses of Dirichlet $L$-functions involving conductors

TL;DR

This paper derives an asymptotic formula for the negative second moment of Dirichlet L-functions involving conductors, with applications to improving bounds in algorithms related to cyclotomic number fields.

Contribution

It provides a new asymptotic formula for negative moments of Dirichlet L-functions and applies it to enhance bounds in algebraic number theory algorithms.

Findings

Asymptotic formula for negative second moment of L(1,χ)

Improved lower bounds on algorithm success probability

Enhanced understanding of Dirichlet L-functions behavior

Abstract

We deal with negative square moments of Dirichlet $L$-functions. Summing over characters modulo $q$, we obtain an asymptotic formula for the negative second moment of $L(1,\chi)$ involving conductors. As an application, we give the improved lower bound on the success probability of the algorithm which recovers a short generator of the input generator of a principal ideal sampled from a specific Gaussian distribution in cyclotomic number fields.