A proof of the Riemann hypothesis

TL;DR

The work embeds the Riemann hypothesis within an adelic trace-formula framework, showing that the trace $Δ(h)$ on $L^2(C_S)$ is nonnegative and can be approximated by $Δ(h_{n,ε})$ which converge to $2\lambda_n$ for Li’s coefficients. By decomposing the trace into contributions from $E_S(Q_Λ^⊥)$ and $E_S(Q_Λ)$ and proving the orthogonal part vanishes while the Λ-part is nonnegative, the authors connect positivity of $λ_n$ to the RH via Li's criterion. A key step is constructing a family of test functions $h_{n,ε}$ with $\widehat{h}_{n,ε}(0)=\widehat{h}_{n,ε}(1)=0$, ensuring $Δ(h_{n,ε})\to 2\lambda_n$. The results offer a novel operator-analytic route to RH grounded in the Connes–Tate adelic framework and trace formulas, with the Li coefficients emerging as trace limits of carefully chosen kernels.

Abstract

In this paper we study traces of an integral operator on two orthogonal subspaces of a $L^2$ space. One of the two traces is shown to be zero. Also, we prove that the trace of the operator on the second subspace is nonnegative. Hence, the operator has a nonnegative trace on the $L^2$ space. This implies the positivity of Li's criterion. By Li's criterion, all nontrivial zeros of the Riemann zeta-function lie on the critical line.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3