Analytic implication from the prime number theorem
Let $x\ge 2$. The $ψ$-form of the prime number theorem is $ψ(x) =\sum\sb{n \le x}Λ(n) =x +O\bigl(x\sp{1-H(x)} \log\sp{2} x\big)$, where $H(x)$ is a certain function of $x$ with $0< H(x) \le \tfrac{1}{2}$. Turán proved in 1950 that this $ψ$-form implies that there are no zeros of $ζ(s)$ 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 Turán'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 Turán's power sum method and using the Lindelöf hypothesis in the zero growth rate form, which is proved recently.