On general prime number theorems with remainder

Abstract

We show that for Beurling generalized numbers the prime number theorem in remainder form (Formula presented.) is equivalent to (for some a > 0) (Formula presented.) where N and π 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 Cesàro sense.