A modified Riemann zeta distribution in the critical strip

Abstract:

Let $\sigma , t \in {\mathbb {R}}$, $s=\sigma +\textrm {{i}}t$ and $\zeta (s)$ be the Riemann zeta function. Put $f_\sigma (t):=\zeta (\sigma -\textrm {{i}}t)/(\sigma -\textrm {{i}}t)$ and $F_\sigma (t):= f_\sigma (t)/f_\sigma (0)$. We show that $F_\sigma (t)$ is a characteristic function of a probability measure for any $0 < \sigma \ne 1$ by giving the probability density function. By using this fact, we show that for any $C \in {\mathbb {C}}$ satisfying $|C| > 10$ and $-19/2 \le \Re C \le 17/2$, the function $\zeta (s) + Cs$ does not vanish in the half-plane $\sigma >1/18$. Moreover, we prove that $F_\sigma (t)$ is an infinitely divisible characteristic function for any $\sigma >1$. Furthermore, we show that the Riemann hypothesis is true if each $F_\sigma (t)$ is an infinitely divisible characteristic function for each $1/2 < \sigma <1$.