prime number theorem

1. mathematicsThe theorem that the number of prime numbers less than n asymptotically approaches n / ln(n) as n approaches infinity.

   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).

   1932, A. E. Ingham, Albert Edward Ingham, The Distribution of Prime Numbers, Cambridge University Press, page 39

   2 more examples

   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.

   2016, Benjamin Fine, Gerhard Rosenberger, Number Theory: An Introduction via the Density of Primes, 2nd edition, Springer (Birkhäuser), page 145

plural prime number theorems

1. mathematicsAny theorem that concerns the distribution of prime numbers.

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

   1996, Illinois Journal of Mathematics, volume 40, University of Illinois Press, page 245