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Explicit quadratic large sieve inequality

Zihao Liu

TL;DR

This work provides an explicit version of Heath-Brown's quadratic large sieve inequality for quadratic Dirichlet characters, delivering precise $\varepsilon$-dependent bounds via an iterated-exponential framework. The authors develop a Heath-Brown–style recursive method, combining Poisson summation, dual sums, and careful control of main terms and errors to obtain a bound of the form $\mathcal{B}(M,N) \le \exp_4(C\varepsilon^{-1}) (MN)^{\varepsilon} (M+N)$ for $MN \ge \exp_4(1)$. They then derive concrete corollaries: an explicit quadratic large sieve inequality, fourth-moment bounds for Dirichlet $L$-functions twisted by quadratic characters, and asymptotics for rational primes splitting in quadratic fields, with quantified exceptional sets. The results hold without any structure on the coefficients $\{a_n\}$, making them broadly applicable to sieve-constructed sequences and enabling unconditional applications in moments and arithmetic statistics. Overall, the paper provides a robust, explicitly quantified baseline for quadratic large-sieve applications, potentially guiding further refinements to reduce the iterated-exponential growth.

Abstract

In this article, we obtain an explicit version of Heath-Brown's large sieve inequality for quadratic characters and discuss its applications to $L$-functions and quadratic fields.

Explicit quadratic large sieve inequality

TL;DR

This work provides an explicit version of Heath-Brown's quadratic large sieve inequality for quadratic Dirichlet characters, delivering precise -dependent bounds via an iterated-exponential framework. The authors develop a Heath-Brown–style recursive method, combining Poisson summation, dual sums, and careful control of main terms and errors to obtain a bound of the form for . They then derive concrete corollaries: an explicit quadratic large sieve inequality, fourth-moment bounds for Dirichlet -functions twisted by quadratic characters, and asymptotics for rational primes splitting in quadratic fields, with quantified exceptional sets. The results hold without any structure on the coefficients , making them broadly applicable to sieve-constructed sequences and enabling unconditional applications in moments and arithmetic statistics. Overall, the paper provides a robust, explicitly quantified baseline for quadratic large-sieve applications, potentially guiding further refinements to reduce the iterated-exponential growth.

Abstract

In this article, we obtain an explicit version of Heath-Brown's large sieve inequality for quadratic characters and discuss its applications to -functions and quadratic fields.
Paper Structure (19 sections, 22 theorems, 150 equations)

This paper contains 19 sections, 22 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Motivations and applications
  3. Statement of results
  4. Organization of the paper
  5. Acknowledgements
  6. Notations
  7. Outline for the proof of \ref{['th-main']}
  8. Preparation for Poisson summation
  9. Asymptotic developments
  10. Recursive estimates
  11. Auxiliary lemmas
  12. Proof of \ref{['th-main']}
  13. Preparation for Poisson summation
  14. Asymptotic developments
  15. Recursive estimates
  16. ...and 4 more sections

Key Result

Theorem 1

Let $M,N\ge1$ and $\{ a_n\}_{n\sim N}$ be a sequence of complex numbers. Define Then there exists a constant $C>0$ such that for every $\varepsilon>0$, we have where $\exp_k$ refers to the k'th iterated exponential.

Theorems & Definitions (43)

  • Theorem 1
  • Corollary 1: Explicit quadratic large sieve inequality
  • Corollary 2: Fourth moment of Dirichlet $L$-functions
  • Remark
  • Corollary 3: Rational primes splitting in $\mathbb Q(\sqrt{-q})$
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark
  • Lemma 4
  • ...and 33 more