An improvement of the large sieve for square moduli under Hooley's hypothesis R* for short Salié sums
Abstract
Let S(Q,M,N,(an))=∑q≤ Q∑abmod q2,(a,q)=1|∑M<n≤ M+Nane(an/q2)|2 be Zhao's large sieve sum with square moduli. Zhao conjectured that S≪(QN)ε(Q3+N)∑|an|2. The bound of Baier and Zhao (2008) exceeds this conjecture by a factor which reaches Q1/2 at the critical point N=Q3, and there the exponent 1/2 has not been lowered since. We prove that, under Hooley's Hypothesis R* for short Salié sums, S≪(QN)ε(Q3+N+Q4/3N2/3)∑|an|2. This replaces the excess over Zhao's conjecture by its 2/3-th power throughout the range Q5/2<N<Q4; at N=Q3 it gives S≪ Q1/2−1/6+εN∑|an|2. The proof is short. Following Zhao's original treatment, we approximate the point under consideration by fractions b/r with denominators up to Q2/3N1/3, which for N<Q4 is beyond the customary √N. If the approximation is poor, an elementary count suffices. If it is good, the analytic part of the problem disappears: after Poisson summation in both variables the count is expressed through quadratic Gauss sums, and their exact evaluation at every modulus turns the sum over the dual variable into short Kloosterman and Salié sums, to which Hypothesis R* applies and gives the square root of the elementary count. At odd moduli the hypothesis is needed only in the form of Conjecture (C*) of Friedlander and Iwaniec. The Gauss-sum evaluations, including even moduli and coefficients sharing a factor with the modulus, are proved in full. This version supersedes earlier versions, which obtained the exponent 1/2−1/28 at N=Q3 by a much longer route through sums of modular square roots. The paper was prepared in collaboration with Claude (Anthropic).
Related
- Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors
- Large values of cubic and quartic Dirichlet L-functions of prime moduli
- Large values of quadratic Hecke L-functions with prime-related moduli of the Gaussian field
- The Weil-Petersson diameter of moduli space for large genus
- On moments of the Erdős-Hooley Delta-function
BibTeX
@misc{baier2026improvement,
title = {{An improvement of the large sieve for square moduli under Hooley's hypothesis $R^*$ for short Salié sums}},
author = {Stephan Baier},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.21195},
eprint = {2609.21195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21195},
}
RIS
TY - UNPB
TI - An improvement of the large sieve for square moduli under Hooley's hypothesis R^* for short Salié sums
AU - Stephan Baier
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2609.21195
UR - https://arxiv.org/abs/2609.21195
AB - Let S(Q,M,N,(a_n))=∑_q≤ Q∑_abmod q^2,(a,q)=1|∑_M<n≤ M+Na_ne(an/q^2)|^2 be Zhao's large sieve sum with square moduli. Zhao conjectured that S≪(QN)^ε(Q^3+N)∑|a_n|^2. The bound of Baier and Zhao (2008) exceeds this conjecture by a factor which reaches Q^1/2 at the critical point N=Q^3, and there the exponent 1/2 has not been lowered since. We prove that, under Hooley's Hypothesis R^* for short Salié sums, S≪(QN)^ε(Q^3+N+Q^4/3N^2/3)∑|a_n|^2. This replaces the excess over Zhao's conjecture by its 2/3-th power throughout the range Q^5/2<N<Q^4; at N=Q^3 it gives S≪ Q^1/2−1/6+εN∑|a_n|^2. The proof is short. Following Zhao's original treatment, we approximate the point under consideration by fractions b/r with denominators up to Q^2/3N^1/3, which for N<Q^4 is beyond the customary √N. If the approximation is poor, an elementary count suffices. If it is good, the analytic part of the problem disappears: after Poisson summation in both variables the count is expressed through quadratic Gauss sums, and their exact evaluation at every modulus turns the sum over the dual variable into short Kloosterman and Salié sums, to which Hypothesis R^* applies and gives the square root of the elementary count. At odd moduli the hypothesis is needed only in the form of Conjecture (C*) of Friedlander and Iwaniec. The Gauss-sum evaluations, including even moduli and coefficients sharing a factor with the modulus, are proved in full. This version supersedes earlier versions, which obtained the exponent 1/2−1/28 at N=Q^3 by a much longer route through sums of modular square roots. The paper was prepared in collaboration with Claude (Anthropic).
ER -