Analytic implications from the remainder term of the prime number theorem

TL;DR

This paper explores the implications of the remainder term in the prime number theorem, linking the distribution of primes to the zeros of the Riemann zeta function and extending Turán's results using power sum methods.

Contribution

It generalizes Turán's results by removing restrictions on the functions related to the zero-free region of the zeta function, using revised power sum techniques.

Findings

Extended the zero-free region implications for the zeta function.

Revised Turán's power sum method for broader applicability.

Connected the remainder term of the prime number theorem to zero distribution.

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.