  We show that for Beurling generalized numbers the prime number theorem in
remainder form $$\pi(x) = \operatorname*{Li}(x) +
O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n\in\mathbb{N}$$ is
equivalent to (for some $a>0$) $$N(x) = ax + O\left(\frac{x}{\log^{n}x}\right)
\quad \mbox{for all } n \in \mathbb{N},$$ where $N$ and $\pi$ are the counting
functions of the generalized integers and primes, respectively. This was
already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on
the subject contains some mistakes. We also obtain an average version of this
prime number theorem with remainders in the Ces\`aro sense.
