Meaning of prime number theorem | Babel Free

Definitions

1. The theorem that the number of prime numbers less than n asymptotically approaches n / ln(n) as n approaches infinity.
2. Any theorem that concerns the distribution of prime numbers.

Equivalents

Examples

“But we cannot infer from them the equivalence in any sense⟳ of these two propositions, since we have⟳ used in our proof of the prime number⟳ theorem a subsidiary theorem on the order⟳ of magnitude of #92;zeta'(s)#47;#92;zeta(s).”

“1974 [Academic Press⟳], Harold M. Edwards, Riemann's Zeta Function, 2001, Dover, page 182, The problem of locating the roots p of ζ, and consequently the problem of estimating the error in the prime number⟳ theorem, is closely related to the problem of estimating the growth of ζ in the critical strip⟳ 0<Res<1 as Ims→∞.”

“In 1859, Riemann attempted to give⟳ a complete⟳ proof of the prime number⟳ theorem using the zeta function for complex variables s. Although he was not successful in proving the prime number⟳ theorem he established many properties of the zeta function and showed that the prime number⟳ theorem depended on the zeros of the zeta function.”

“In [6], [11], abstract prime number⟳ theorems are proved under a variety of conditions.”

CEFR level