Year: 2010
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Cheng, Yuanyou, Fox, Glenn, Hassani, Mehdi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1484080680