On General Prime Number Theorems with Remainder

  title={On General Prime Number Theorems with Remainder},
  author={Gregory Debruyne and Jasson Vindas},
  • Gregory Debruyne, Jasson Vindas
  • Published 2017
  • Mathematics
  • 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… CONTINUE READING

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