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Short Exponential Sums and Ternary Correlations of Multiplicative Functions

Jiseong Kim

Abstract

Let $f_1, f_2, f_3$ be $k$-divisor-bounded functions, at least one of which satisfies certain second-moment integral bounds. We show that for any $\varepsilon > 0$ and \[ X^{1/2+100\varepsilon} \ll H \ll X^{1-\varepsilon}, \] we have \[ \sum_{|h|\le H} \left(1-\frac{|h|}{H}\right) \sum_{X \le n \le 2X} f_{1}(n)\, f_{2}(n+h)\, f_{3}(n+2h) = O\!\left(XH^{1-\varepsilon}\right). \] Our approach differs from previous methods based on spectral theory or Heath-Brown-type decompositions, and instead combines the circle method with weighted short exponential-sum bounds. The key input is short exponential-sum estimates obtained from integral moment bounds for $L$-functions.

Short Exponential Sums and Ternary Correlations of Multiplicative Functions

Abstract

Let be -divisor-bounded functions, at least one of which satisfies certain second-moment integral bounds. We show that for any and we have Our approach differs from previous methods based on spectral theory or Heath-Brown-type decompositions, and instead combines the circle method with weighted short exponential-sum bounds. The key input is short exponential-sum estimates obtained from integral moment bounds for -functions.
Paper Structure (16 sections, 10 theorems, 108 equations, 1 table)

This paper contains 16 sections, 10 theorems, 108 equations, 1 table.

Table of Contents

  1. Introduction
  2. Short exponential sum estimates
  3. Ternary correlations
  4. Corollaries
  5. Non-vanishing over three term arithmetic progressions
  6. Sketch of the proof
  7. Notation
  8. Plan of the paper
  9. Exponential Sums over Short Intervals
  10. Proof of Theorem \ref{['Theorem']}
  11. Major arcs
  12. The main term
  13. Minor arcs
  14. Proof of Theorem \ref{['Theorem2']}
  15. Bounding short sums on the major arcs
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $g \in \mathcal{F}_k(0)$, and let $0 \le a < q < X$. Let $x \in [X,2X]$, and let $0 < \eta < 1$. Then, for any $|\gamma|<1$ with $|\gamma| H^{\eta - \varepsilon/2} \to \infty$, we have

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 10 more