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A large sieve inequality for characters to quadratic moduli

C. C. Corrigan

TL;DR

This work proves a large sieve inequality for additive characters with moduli given by a quadratic polynomial $f$, extending known results from square moduli to general degree-two polynomials. The core is a dyadic, Farey-based reduction together with a principal estimate bounding a key quantity $K_\delta$ via a sophisticated exponential-sum analysis that uses Gauss sums and short-interval counting. The main outcome yields a bound for $\mathscr{N}_f(Q,N)$ that sharpens the trivial estimate in the range $f(Q)\ll N\ll f(Q)^2$ and confirms the conjectural behavior in $f(Q)\ll N\ll f(Q)\sqrt{Q}$, while enabling a weighted zero-density result for twists of Dirichlet $L$-functions by characters with conductors of the form $f(q)$. The paper further derives a multiplicative analogue and discusses the implications for zero distribution and mean values of Dirichlet polynomials, including conditional improvements under conjectures for higher moments.

Abstract

In this article, we establish a large sieve inequality for additive characters to moduli in the range of appropriate integer polynomials of degree two. As an application, we derive a weighted zero-density estimate for twists of $L$-functions associated to multiplicative characters with conductor of this form.

A large sieve inequality for characters to quadratic moduli

TL;DR

This work proves a large sieve inequality for additive characters with moduli given by a quadratic polynomial , extending known results from square moduli to general degree-two polynomials. The core is a dyadic, Farey-based reduction together with a principal estimate bounding a key quantity via a sophisticated exponential-sum analysis that uses Gauss sums and short-interval counting. The main outcome yields a bound for that sharpens the trivial estimate in the range and confirms the conjectural behavior in , while enabling a weighted zero-density result for twists of Dirichlet -functions by characters with conductors of the form . The paper further derives a multiplicative analogue and discusses the implications for zero distribution and mean values of Dirichlet polynomials, including conditional improvements under conjectures for higher moments.

Abstract

In this article, we establish a large sieve inequality for additive characters to moduli in the range of appropriate integer polynomials of degree two. As an application, we derive a weighted zero-density estimate for twists of -functions associated to multiplicative characters with conductor of this form.
Paper Structure (7 sections, 21 theorems, 108 equations)

This paper contains 7 sections, 21 theorems, 108 equations.

Table of Contents

  1. Introduction.
  2. The strategy.
  3. Preliminary lemmata.
  4. Two preliminary estimates.
  5. The principal estimate.
  6. Establishment of the main assertions.
  7. Demonstration of the corollary.

Key Result

Theorem 1.1

Suppose that $f:\boldsymbol{\mathrm{N}}\hookrightarrow\boldsymbol{\mathrm{N}}$ is a strictly increasing polynomial of degree two, with leading coefficient $A$ and discriminant $\Delta_f$. Then, for any $N\geqslant3$, if $Q\geqslant|\Delta_f|/A^2$ is sufficiently large that $3AQ^2\geqslant f(Q)$, the where the implied constant depends on $\varepsilon$ alone.

Theorems & Definitions (36)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • ...and 26 more