Analytic implications from the remainder term of the prime number theorem

Abstract

It is well known that the distribution of the prime numbers plays a central role in number theory. It has been known, since Riemann's memoir in 1860, that the distribution of prime numbers can be described by the zero-free region of the Riemann zeta function $\zeta(s)$. This function has infinitely many zeros and a unique pole at $s = 1$. Those zeros at $s = -2, -4, -6, ...$ are known as trivial zeros. The nontrivial zeros of $\zeta(s)$ are all located in the so-called critical strip $0<\Re(s) <1$. Define $\Lambda(n) =\log p$ whenever $n =p\sp{m}$ for a prime number $p$ and a positive integer $m$, and zero otherwise. 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 the sum runs through the set of positive integers and $H(x)$ is a certain function of $x$ with $\tfrac{1}{2} \le H(x) < 1$. 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 connected to $H(x)$ in a certain way with $1/2 \le h(t) <1$ but both $H(x)$ and $h(t)$ are close to 1. We prove results similar to Tur\'an's where $\tfrac{1}{2} \le H(x)< 1$ and $\tfrac{1}{2} \le h(t) <1$ in altered forms without any other restrictions. The proof involves slightly revising and applying Tur\'an's power sum method.