The large sieve for self-dual Eisenstein series of varying levels

TL;DR

This work proves an essentially optimal large sieve inequality for self-dual Eisenstein series with varying levels, interpretable as a large sieve for rationals ordered by height. The authors develop a novel recursive strategy that blends a Dirichlet L-function functional equation (FE) approach with a divisor-switching family-average method, drawing connections to Heath-Brown’s quadratic sieve and the asymptotic large sieve of Conrey–Iwaniec–Soundararajan. The main bound, $\Delta(Q,k,T,N) \ll_\varepsilon (Q^2 k T + N) (Q k T N)^{\varepsilon}$, is complemented by an additive variant and rational-height implications, showcasing sharp control over GL2-type families with varying nebentypus. Applications include rational-height sieve problems and a Barban–Davenport–Halberstam type theorem, with potential for broader rational large-sieve phenomena and further arithmetic consequences.

Abstract

We prove an essentially optimal large sieve inequality for self-dual Eisenstein series of varying levels. This bound can alternatively be interpreted as a large sieve inequality for rationals ordered by height. The method of proof is recursive, and has some elements in common with Heath-Brown's quadratic large sieve, and the asymptotic large sieve of Conrey, Iwaniec, and Soundararajan.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5: Recursive functional equation
• Theorem 1.6: Recursive family average
• Proposition 1.7
• Proposition 1.8
• Lemma 2.1
• proof