Analytic implications from the remainder term of the prime number theorem

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 $s=-2,-4,-6,\mathrm{.}\mathrm{.}\mathrm{.}$ are known as trivial zeros. The nontrivial zeros of $\zeta (s)$ are all located in the so-called critical strip $0<\mathrm{\Re }(s)<1$. Define $\mathrm{\Lambda }(n)=\log p$ whenever $n =p\sp{m}$foraprimenumber$p$andapositiveinteger$m$, and zero otherwise. Let $x\ge 2$. The $\psi$-form of the prime number theorem is $\psi(x) =\sum\sb{n \le x}\Lambda(n) =x +O\bigl(x\sp{1-H(x)} \log\sp{2} x\big)$,wherethesumrunsthroughthesetofpositiveintegersand$H(x)$ is a certain function of $x$ with $\frac{1}{2}\le H(x)<1$. Tur'an proved in 1950 that this $\psi$-form implies that there are no zeros of $\zeta (s)$ for $\mathrm{\Re }(s)>h(t)$, where $t=\mathrm{\Im }(s)$, and $h(t)$ is a function connected to $H(x)$ in a certain way with $1\mathrm{/}2\le h(t)<1$ but both $H(x)$ and $h(t)$ are close to

1. We prove results similar to Tur'an’s where $\frac{1}{2}\le H(x)<1$ and $\frac{1}{2}\le h(t)<1$ in altered forms without any other restrictions. The proof involves slightly revising and applying Tur'an’s power sum method.