The Prime Number Theorem

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01 octobre 1997

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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

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Publié par

sucun

Publié le

01 octobre 1997

Nombre de lectures

Langue

English

The Prime Number Theorem for Arithmetic Progressions

For the origin of this text, see Kedlaya’s “Primes in Arithmetic Progressions” http://www-math.mit.edu/ ∼ kedlaya/18.785/dirichlet.pdf, available at http://www-math.mit.edu/ ∼ kedlaya/18.785. Serre’s Coursd’arithm´etique . Soprunov’s “A Short Proof of the Prime Number Theorem for Arithmetic Pro-gressions” http://www.math.umass.edu/ ∼ isoprou/pdf/primes.pdf, available at http://www.math.umass.edu/ ∼ isoprou/research.html. 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 http://maths.dur.ac.uk/ ∼ dma0hg/prime number theorem zagier.pdf http://www.math.sjsu.edu/ ∼ goldston/zagierPNT.pdf http://mathdl.maa.org/images/upload library/22/Chauvenet/Zagier.pdf. Let m be a nonzero integer, let a an integer prime to m , let π a ( x ) be the number of those primes which are congruent to a mod m and which do not exceed x , and let ϕ 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 :

π a ( x ) ∼ ϕ l x og 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 /m Z , let χ be a character of G ,

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