Notes on $\log (\zeta (s))^\prime \prime $

Abstract

Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann $\zeta $ function, in particular, the zeros of this function. Theorem~1.2 gives a zero-free region. Theorem~1.4 gives an asymptotic estimate for the number of nontrivial zeros to height $T$. Theorem~1.7 is a zero density estimate.