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Bilinear sums with $GL(2)$ coefficients and the exponent of distribution of $d_3$

Prahlad Sharma

Abstract

We obtain the exponent of distribution $1/2+1/30$ for the ternary divisor function $d_3$ to square-free and prime power moduli, improving the previous results of Fouvry--Kowalski--Michel, Heath-Brown, and Friedlander--Iwaniec. The key input is certain estimates on bilinear sums with $GL(2)$ coefficients obtained using the delta symbol approach.

Bilinear sums with $GL(2)$ coefficients and the exponent of distribution of $d_3$

Abstract

We obtain the exponent of distribution for the ternary divisor function to square-free and prime power moduli, improving the previous results of Fouvry--Kowalski--Michel, Heath-Brown, and Friedlander--Iwaniec. The key input is certain estimates on bilinear sums with coefficients obtained using the delta symbol approach.
Paper Structure (23 sections, 12 theorems, 267 equations)

This paper contains 23 sections, 12 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Voronoi summation formula for $d_3(n)$.
  4. Voronoi summation formula for $GL(2)$.
  5. Character sum estimates
  6. Proof of Theorem \ref{['prime']}
  7. Dualisation
  8. Simplifying the character sum
  9. Cauchy-Schwarz and Poisson
  10. The zero frequency
  11. Non-zero frequencies
  12. Optimal choice for $L$
  13. Proof of Theorem \ref{['primepower']}
  14. Dualisation
  15. Simplifying the character sum
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\epsilon>0$ and $a$ be a non-zero integer. For every square-free $q\geq 1$ and every odd prime power $q=p^{\gamma}, \gamma\geq 28$ with $(a,q)=1$ and satisfying we have where the implied constant depends only on $\epsilon$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • remark 1: Notation
  • Lemma 2.1: X. Li
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 10 more