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

Iu-Iong Ng, Yuichiro Toma

TL;DR

The paper derives asymptotic formulas for negative moments of Dirichlet L-functions at $s=1$, including a conductor-weighted variant, by combining contour integration with auxiliary lemmas and Dirichlet character orthogonality. It achieves a GRH-free asymptotic for the negative second moment and, for prime-power moduli $q=p^k$, a conductor-weighted negative square moment with explicit main terms and quantitative error terms under the nonexistence of exceptional zeros in key cases. The results are then applied to the Short Generator Problem in cyclotomic number fields, bounding the lengths of dual log-unit bases and improving the success probability of short-generator recovery when the input is randomly distributed as a Gaussian in the cyclotomic unit lattice. Collectively, these findings connect negative $L$-value moments to cryptographic performance and provide unconditional progress (in the $k=2$ case) alongside refined conductor-dependent estimates with practical implications for SGP in cyclotomic fields.

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 $L(1,χ)$ 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.

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

TL;DR

The paper derives asymptotic formulas for negative moments of Dirichlet L-functions at , including a conductor-weighted variant, by combining contour integration with auxiliary lemmas and Dirichlet character orthogonality. It achieves a GRH-free asymptotic for the negative second moment and, for prime-power moduli , a conductor-weighted negative square moment with explicit main terms and quantitative error terms under the nonexistence of exceptional zeros in key cases. The results are then applied to the Short Generator Problem in cyclotomic number fields, bounding the lengths of dual log-unit bases and improving the success probability of short-generator recovery when the input is randomly distributed as a Gaussian in the cyclotomic unit lattice. Collectively, these findings connect negative -value moments to cryptographic performance and provide unconditional progress (in the case) alongside refined conductor-dependent estimates with practical implications for SGP in cyclotomic fields.

Abstract

We deal with negative square moments of Dirichlet -functions. Summing over characters modulo , we obtain an asymptotic formula for the negative second moment of 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.
Paper Structure (10 sections, 16 theorems, 92 equations, 1 table)

This paper contains 10 sections, 16 theorems, 92 equations, 1 table.

Table of Contents

  1. Introduction
  2. Negative moment
  3. Main results
  4. Auxiliary lemmas
  5. Proof of Theorem \ref{['thm:negative square moment']}
  6. Proof of Theorem \ref{['thm:negative square moment with conductor']}
  7. Application in recovering short generators
  8. Previous results on SGP over cyclotomic number fields
  9. Our improvements
  10. Proof Theorem \ref{['thm:CDPR_main']}

Key Result

Theorem 1.2

Let $q$ be a positive integer and $\chi$ be a Dirichlet character modulo $q$. Then we have for an absolute constant $C>0$, where $\beta_1$ denotes the exceptional zero (defined in Proposition pro:exceptional zero), and $\delta_1=1$ if $\beta_1$ exists, or $\delta_1=0$ otherwise.

Theorems & Definitions (27)

  • Conjecture 1.1: Gonek
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 17 more