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Erschienen in:
2017 | OriginalPaper | Buchkapitel
On General Prime Number Theorems with Remainder
verfasst von : Gregory Debruyne, Jasson Vindas
Erschienen in: Generalized Functions and Fourier Analysis
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We show that for Beurling generalized numbers the prime number theorem in remainder form $$ \pi \left( x \right) = Li\left( x \right) + O\left( {\frac{x} {{\log ^n x}}} \right)\,for\,all\,n\, \in \,{\Bbb N} $$ is equivalent to (for some a > 0) $$ N\left( x \right) = ax + O\left( {\frac{x} {{\log ^n x}}} \right)\,for\,all\,n\, \in \,{\Bbb N} $$ 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.