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Fourier optimization and consequences of the generalized Riemann hypothesis

Emily Quesada-Herrera

Abstract

We give an exposition of some connections between Fourier optimization problems and problems in number theory. In particular, we present some recent conditional bounds under the generalized Riemann hypothesis, achieved via a Fourier optimization framework, on bounding the maximum possible gap between consecutive prime numbers represented by a given quadratic form; and on bounding the least quadratic non-residue modulo a prime number. This is based on joint works with Emanuel Carneiro, Andrés Chirre, Micah Milinovich, and Antonio Pedro Ramos.

Fourier optimization and consequences of the generalized Riemann hypothesis

Abstract

We give an exposition of some connections between Fourier optimization problems and problems in number theory. In particular, we present some recent conditional bounds under the generalized Riemann hypothesis, achieved via a Fourier optimization framework, on bounding the maximum possible gap between consecutive prime numbers represented by a given quadratic form; and on bounding the least quadratic non-residue modulo a prime number. This is based on joint works with Emanuel Carneiro, Andrés Chirre, Micah Milinovich, and Antonio Pedro Ramos.

Paper Structure

This paper contains 7 sections, 5 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fourier transform
  3. Riemann zeta-function and Dirichlet $L$-functions
  4. A link between worlds
  5. Prime gaps and quadratic forms
  6. Least quadratic non-residue
  7. Numerically optimizing the bounds

Key Result

Theorem 1

Assume the generalized Riemann hypothesis. Then, for all sufficiently large $x$, there is always a prime number $p$ of the form $p=u^2+27v^2$ between $x$ and $x+\frac{46}{25}\sqrt{x}\log x$.

Figures (1)

  • Figure 1: Plot of $\widehat{F}(t)/\widehat{F}(0)$ defined in \ref{['eq:F']}, a good test function for the optimization problem \ref{['eq:prob2']}.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Lemma 3: Guinand-Weil explicit formula
  • Theorem 4
  • Theorem 5