A generalization of the prime number theorem

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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:  [image omitted], where F:  →  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.
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