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On bilinear sums with modular square roots and applications II

Stephan Baier

Abstract

In this article, we continue our recent investigations on bilinear sums and additive energies with modular square roots. Here we improve our recent results for the case when the ranges of variables are large. We use these results to make further partial progress on the large sieve for square moduli.

On bilinear sums with modular square roots and applications II

Abstract

In this article, we continue our recent investigations on bilinear sums and additive energies with modular square roots. Here we improve our recent results for the case when the ranges of variables are large. We use these results to make further partial progress on the large sieve for square moduli.
Paper Structure (24 sections, 15 theorems, 194 equations)

This paper contains 24 sections, 15 theorems, 194 equations.

Table of Contents

  1. Introduction and main results
  2. Notations
  3. Summary of this article
  4. Bilinear sums with modular square roots
  5. Large sieve with square moduli
  6. Preliminaries
  7. Bilinear sums
  8. Weyl differencing and Poisson summation
  9. Smoothing
  10. Evaluation of quadratic Gauss sums
  11. Reduction to complete exponential sums
  12. Multiplicativity of our exponential sums
  13. Estimation of our exponential sums
  14. Setup
  15. The case $m=1$
  16. ...and 9 more sections

Key Result

Theorem 1

(Baier) Suppose that $r,j\in \mathbb{N}$, $(r,j)=1$ and $1\leqslant L,M\leqslant r$. Let $f:[1,M]\rightarrow \mathbb{R}$ be a continuously differentiable function such that $|f'(x)|\leqslant F$ on $[1,M]$, where $F\leqslant L^{-1}$. Let $\boldsymbol{\alpha}=(\alpha_l)_{|l|\leqslant L}$ and $\boldsym Then we have where

Theorems & Definitions (23)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Proposition 1: Weyl differencing
  • proof
  • Proposition 2: Poisson summation
  • proof
  • Proposition 3
  • ...and 13 more