The Prime Number Theorem

Niveau: Secondaire, Lycée
The Prime Number Theorem for Arithmetic Progressions For the origin of this text, see Kedlaya's “Primes in Arithmetic Progressions” available at Serre's Cours d'arithmetique. Soprunov's “A Short Proof of the Prime Number Theorem for Arithmetic Pro- gressions” available at Zagier's “Newman's Short Proof of the Prime Number Theorem”, The American Mathematical Monthly, Vol. 104, No. 8 (Oct. 1997), pp. 705-708, available at number theorem zagier.pdf library/22/Chauvenet/Zagier.pdf. Letm be a nonzero integer, let a an integer prime tom, let pia(x) be the number of those primes which are congruent to a mod m and which do not exceed x, and let ? be the number of those positive integers which are prime to m and less than m. Recall that f(x) ? g(x) means that f/g tends to 1. We wish to prove the famous Prime Number Theorem for Arithmetic Pro- gressions: pia(x) ? x ? log x . (1) Recall that a character of a finite abelian group G is a morphism from G into the circle group, and that the characters of G form a finite multiplicative abelian group G?. Let G be again the group of units of Z/mZ, let ? be a character of G, 1

• zeta function


• group


• write again ?


• any finite abelian


• arithmetic pro- gressions


• dt ≤ ∫


• riemann's zeta


• dt