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A proof of Riemann Hypothesis

Pengcheng Niu, Junli Zhang

Abstract

Let $Ξ(t)$ be a function relating to the Riemann zeta function $ζ(s)$ with $s = \frac{1} {2} + it$. In this paper, we construct a function $v$ containing $t$ and $Ξ(t)$, and prove that $v$ satisfies a nonadjoint boundary value problem to a nonsingular differential equation if $t$ is any nontrivial zero of $Ξ(t)$. Inspecting properties of $v$ and using known results of nontrivial zeros of $ζ(s)$, we derive that nontrivial zeros of $ζ(s)$ all have real part equal to $\frac{1} {2}$, which concludes that Riemann Hypothesis is true.

A proof of Riemann Hypothesis

Abstract

Let be a function relating to the Riemann zeta function with . In this paper, we construct a function containing and , and prove that satisfies a nonadjoint boundary value problem to a nonsingular differential equation if is any nontrivial zero of . Inspecting properties of and using known results of nontrivial zeros of , we derive that nontrivial zeros of all have real part equal to , which concludes that Riemann Hypothesis is true.

Paper Structure

This paper contains 7 sections, 96 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. The construction of function $v$
  4. The imaginary part $f(b;\varepsilon )$ of $P(b;\varepsilon )$
  5. Determining ${b_0}$ and ${\varepsilon _0}$ satisfying (\ref{['eq2.14']}) and (\ref{['eq2.15']})
  6. Finishing the proof of Theorem 1
  7. Proofs of (\ref{['eq2.25']})-(\ref{['eq2.27']})

Theorems & Definitions (1)

  • Remark \oldthetheorem