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On Salem's Integral Equation and related criteria

Alexander E Patkowski

Abstract

We extend Salem's Integral equation to the non-homogenous form, and offer the associated criteria for the Riemann Hypothesis. Explicit solutions for the non-homogenous case are given, which in turn give further insight into Salem's criteria for the RH. As a conclusion, we show these results follow from a corollary relating the uniqueness of solutions of the non-homogenous form with Wiener's theorem.

On Salem's Integral Equation and related criteria

Abstract

We extend Salem's Integral equation to the non-homogenous form, and offer the associated criteria for the Riemann Hypothesis. Explicit solutions for the non-homogenous case are given, which in turn give further insight into Salem's criteria for the RH. As a conclusion, we show these results follow from a corollary relating the uniqueness of solutions of the non-homogenous form with Wiener's theorem.

Paper Structure

This paper contains 3 sections, 5 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorems
  3. Observations and Concluding remarks

Key Result

Theorem 1.1

Suppose that $h(y)\in L_2(\mathbb{R}),$ and $H(x)$ is the Fourier transform of $h(y).$ Then the Riemann hypothesis is equivalent to the claim that has a unique solution for each $h(x)$ in the region $\frac{1}{2}<\sigma<1.$ In particular, provided that

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['thm:thm1']}
  • proof : Proof of Theorem \ref{['thm:thm2']}
  • proof : Proof of Theorem \ref{['thm:thm3']}
  • Corollary 3.0.1
  • proof