  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.
