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On bilinear sums with modular square roots and applications III

Stephan Baier

Abstract

We continue our investigations of bilinear sums with modular square roots and the large sieve for square moduli in our recent article "On bilinear sums with modular square roots and applications II", arXiv:2603.00768. In the present article, we focus on the case of prime square moduli for which our previous method in the said article did not yield any improvement. Now we modify this method to make progress for these moduli. The key idea is to restrict certain quadratic Gauss sums to reduced residue classes, which results in significant cancellations in certain cases.

On bilinear sums with modular square roots and applications III

Abstract

We continue our investigations of bilinear sums with modular square roots and the large sieve for square moduli in our recent article "On bilinear sums with modular square roots and applications II", arXiv:2603.00768. In the present article, we focus on the case of prime square moduli for which our previous method in the said article did not yield any improvement. Now we modify this method to make progress for these moduli. The key idea is to restrict certain quadratic Gauss sums to reduced residue classes, which results in significant cancellations in certain cases.

Paper Structure

This paper contains 14 sections, 12 theorems, 131 equations.

Table of Contents

  1. Introduction and main results
  2. Notations
  3. Previous and new results
  4. Preliminaries
  5. Bilinear sums
  6. Basic approach
  7. Smoothing
  8. Evaluation of quadratic Gauss sums
  9. Estimation of exponential sums
  10. Completion of the proof of Theorem \ref{['mainresult1']}
  11. Large sieve with square moduli
  12. Relation to bilinear sums with modular square roots
  13. Estimation of $P(x)$
  14. Completion of the proof of Theorem \ref{['mainresult2']}

Key Result

Theorem 1

(Baier2) Suppose that $r,j\in \mathbb{N}$, $(r,j)=1$, $1\leqslant L\leqslant r$ and $1\leqslant M\leqslant r/2$. 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 Then we have where In particular, if $r$ is squarefree, then

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Proposition 1: Poisson summation
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 9 more