Analytic implication from the prime number theorem

Abstract

Let $x\ge 2$. The $\psi$-form of the prime number theorem is $\psi(x) =\sum\sb{n \le x}\Lambda(n) =x +O\bigl(x\sp{1-H(x)} \log\sp{2} x\big)$, where $H(x)$ is a certain function of $x$ with $0< H(x) \le \tfrac{1}{2}$. Tur\'an proved in 1950 that this $\psi$-form implies that there are no zeros of $\zeta(s)$ for $\Re(s) > h(t)$, where $t=\Im(s)$, and $h(t)$ is a function related to $H(x)$ with $0< h(t) \le \tfrac{1}{2}$, but both $H(x)$ and $h(t)$ are very close to 1. We prove results similar to Tur\'an's, with $H(x)$ and $ h(t)$ in some altered forms without the restriction that $H(x)$ and $h(t)$ are close to 1. The proof involves slightly revising and applying Tur\'an's power sum method and using the Lindel\"of hypothesis in the zero growth rate form, which is proved recently.