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Decoupling, exponential sums and the Riemann zeta function

Jean Bourgain

Abstract

We establish a new decoupling inequality for curves in the spirit of [B-D1], [B-D2] which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in [H]. In particular, this leads to an improved bound $|ζ(\frac 12+it)|\ll t^{53/342+\varepsilon}$ for the zeta function on the critical line

Decoupling, exponential sums and the Riemann zeta function

Abstract

We establish a new decoupling inequality for curves in the spirit of [B-D1], [B-D2] which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in [H]. In particular, this leads to an improved bound for the zeta function on the critical line

Paper Structure

This paper contains 6 sections, 3 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. A decoupling inequality for curves
  3. A mean value theorem
  4. Applications to exponential sums
  5. Bounding the zeta-function on the critical line
  6. Further comments

Key Result

Theorem 1

Let $\Gamma$ be as above and $I_1, \ldots, I_{\frac{d}{2}}\subset [0, 1]$ subintervals that are $O(1)$-separated, let $N$ be large and $\{I_\tau\}$ a partition of $[0, 1]$ in $N^{-\frac{1}{2}}$-intervals. Then for arbitrary coefficient functions $a_j = a_j (t)$ holds, with $\varepsilon>0$ arbitrary.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • proof
  • Corollary 3