The Hilbert space basis and Hilbert's eighth problem

TL;DR

This paper explores the structure of Hilbert spaces of real functions and demonstrates that the Hardy function related to the Riemann zeta function has only simple, real zeros, linking Hilbert space theory to the Riemann hypothesis.

Contribution

It establishes the existence of a complete orthonormal basis in a Hilbert space that implies the Hardy function's zeros are simple and real, offering insights into the Riemann hypothesis.

Findings

Existence of a complete basis in Hilbert space for Hardy functions

Zeros of the Hardy function are simple and real

Connection between Hilbert space basis and Riemann hypothesis

Abstract

The paper considers the Hilbert space $\hat{H}_r$ of real functions summable with the square $L^2(a,b)_r$ on any interval $\{(a,b)_r\}_{r=1}^{\infty}\in \mathbb{R}$. It is shown on the basis of the theorem on zeros of real orthogonal polynomials if in $\hat{H}_r$ there exists a complete orthonormal basis $\{f(x)_k\}_{k=1}^{\infty}$ and the function $f(x)\in\{f(x)_k\}_{k=1}^{\infty}$ has zeros, then these zeros are simple and real. The generalized Hardy function $Z(\sigma,t)=\Re\zeta(\sigma+it)e^{i\theta(t)}$ is considered. It is shown that in the Hilbert space $\hat{H}_r$ there exists a complete basis $\{Z(\lambda_k,t\}_{k=1}^{\infty}$ where $\lambda_k\in\mathbb{Q}$ and $Z(t)\in\{Z(\lambda_k,t\}_{k=1}^{\infty}$ when $\lambda_k=1/2$, hence the Hardy function $Z(t)=\zeta(1/2+it)e^{i\theta(t)}$ has all simple and real zeros.