On General Prime Number Theorems with Remainder

We show that for Beurling generalized numbers the prime number theorem in remainder form $$\pi(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right),$$ for all $n\in\mathbb{N}, is equivalent to (for some $a>0$) $$N(x) = ax + O\left(\frac{x}{\log^{n}x}\right),$$ 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 Cesaro sense.

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