Analytic implication from the prime number theorem

Abstract

Let $x\ge 2$. The $\psi$-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 $0<H(x)\le \frac{1}{2}$. Turán 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 $0<h(t)\le \frac{1}{2}$, 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.

22 pages, to be submitted to Transaction of the AMS. A slightly revising is needed in the final process. arXiv admin note: text overlap with arXiv:1003.0098

References in corpus (1)

• Proof of the the Riemann hypothesis from the density and Lindelof hypotheses via a power sum method