The large sieve for square moduli under Hooley's hypothesis $R^{\ast }$

Abstract

Let $S(Q,M,N,(a_n)):={\sum }_{q\le Q}{\sum }_{(a,q)=1}|{\sum }_{M<n\le M+N}{a}_{n}e(an/q^2){|}^2$ be Zhao's large sieve sum with square moduli. At the critical point $N=Q^3$ the best known unconditional bound, due to Baier and Zhao (2008), is $S\ll Q^{1/2+\epsilon }N\sum |a_n{|}^2$, against the conjectured $Q^{\epsilon }N\sum |a_n{|}^2$, and the exponent $\frac{1}{2}$ has not been lowered since. We prove that, under Hooley's Hypothesis $R^{\ast }$ for short Salié sums -- square-root cancellation for ${\sum }_{x_1<n\le x_2}\left(\frac{n}{c}\right)e_c(a\bar{n}+bn)$ over arbitrary subintervals of a period -- one has $S\ll Q^{1/2-1/134+\epsilon }N\sum |a_n{|}^2$ at $N=Q^3$. The key estimate is a bound for the number $P(\alpha )$ of fractions $a/q^2$, $q\le Q$, within $Q^{-3}$ of a point $\alpha$ near $b/r$: we show $P(b/r+z)\ll (Q^{2/3}{r}^{-1/3}+Q^{1/4})Q^{\epsilon }$ for every modulus $Q^{1/2+\epsilon }\le r\le Q^{3/2}$, improving the bound $Q^{9/16}{r}^{-1/8}$ obtained by Baier (2026) for $r=p,p^2$ only, and reaching every modulus. The proof rests on a single observation: a sum of modular square roots ${\sum }_{n\in J}{e}_r(a\sqrt{jn})$ over an interval $J$ is, after completion and an exact evaluation of quadratic Gauss sums at every modulus, $r^{-1/2}$ times a Salié sum of length $r/|J|$. Hypothesis $R^{\ast }$ therefore yields square-root cancellation for these sums directly, at every modulus, without Weyl differencing; the saving over the trivial bound is the square of what the Weyl-differencing route gives. The Gauss-sum evaluations, including even moduli and coefficients sharing a factor with the modulus, are proved in full. The paper was prepared in collaboration with Claude (Anthropic); Section 1.9 sets out what each of us contributed.

30 pages