On the Riemann zeta function, II: operators and their indices

Abstract

On $\\Omega=\\{s:\\Re(s)\\ge\\frac12\\}$，$\\frac{s-1}{s^2}\\zeta(s)$ is a bounded analytic function. As a multiplication operator on the Hardy space $H^2(\\Omega)$, its index vanishes if and only if the Riemann hypothesis holds. Through the (inverse) KS-transform, an equivalent statement is true for certain convolution operator on the Hilbert space $L^2([1,\\infty))$. A discrete formulation of such result says that the operator $A_\\zeta=(\\frac1{mn}\\{\\frac{m}{n+1}\\})_{m,n\\ge1}$ has vanishing index on $l^2(\\bN)$ if and only if the Riemann hypothesis is true. Similar results hold for Dirichlet $L$-functions and corresponding generalized Riemann hypothesis. Detailed proofs are given.