One-level density of zeros of the Riemann zeta function weighted by its special values

Source abstract

We consider the one-level density of zeros of the Riemann zeta function weighted by its special values on the line ${\rm Re}(s)=σ$, where $σ$ is a fixed real number such that $1/2\le σ\le1$. We show that the density function for the Riemann zeta function changes if and only if $σ=1/2$.