A generalization of the prime number theorem

Abstract

A general theorem is introduced and proved, a special case of which is the prime number theorem (PNT). The main theorem proved in this article is the following: , where F: ℝ → ℝ is analytic for all x ≥ 2, and the p's in the sum represent all primes p ≤ N. Since PNT is used to derive our generalization, we may say that PNT and our generalization are equivalent.

Keywords:

• prime number theorem (PNT)
• logarithmic integral
• Maclaurin series
• generalized prime number theorem