The Hilbert space basis and Hilbert's eighth problem

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.