Dirichlet theorem on prime numbers in arithmetic progression

Dirichlet’s prime number theorem in arithmetic progression states:

Let be $l,k>0$ ${\ displaystyle l, k> 0}$ ${\ displaystyle l, k> 0}$ Are integers, and $(l,k)=1$ ${\ displaystyle (l, k) = 1}$ ${\ displaystyle (l, k) = 1}$ .

Then there are infinitely many primes $p$ ${\ displaystyle p}$ $p$ such that $p\equiv l{\pmod {k}}$ ${\ displaystyle p \ equiv l {\ pmod {k}}}$ ${\ displaystyle p \ equiv l {\ pmod {k}}}$ .

It follows from this that every infinite arithmetic progression , the first term and the difference of which are natural coprime numbers , contains an infinite number of primes.

Content

Evidence History

The theorem in this formulation was proved by Dirichlet by analytic means in 1837. Later, the proof of the theorem was found by elementary methods ^[1] . Various such evidence have been provided by Mertens, Selberg, and Zassenhaus.

Variations

When considering simple $p\equiv l{\pmod {k}}$ ${\ displaystyle p \ equiv l {\ pmod {k}}}$ quite often it turns out that their set has many properties inherent in the set of all primes. There are many theorems and hypotheses that consider only primes from a certain class of residues, or the relations of sets of primes from different classes of residues.

For example, in addition to the main assertion of Dirichlet ’s theorem, he proved in 1839 that for any fixed natural coprime numbers $l$ ${\ displaystyle l}$ and $k$ ${\ displaystyle k}$ :

\lim _{s\to 1+}{\frac {\sum \limits _{p}{\dfrac {1}{p^{s}}}}{\ln {\dfrac {1}{s-1}}}}={\frac {1}{\varphi (k)}},

{\ displaystyle \ lim _ {s \ to 1 +} {\ frac {\ sum \ limits _ {p} {\ dfrac {1} {p ^ {s}}}} {\ ln {\ dfrac {1} { s-1}}}} = {\ frac {1} {\ varphi (k)}},}

where the summation is over all primes $p$ ${\ displaystyle p}$ with the condition $p\equiv l{\pmod {k}}$ ${\ displaystyle p \ equiv l {\ pmod {k}}}$ , but $\varphi$ ${\ displaystyle \ varphi}$ - Euler function .

This relation can be interpreted as the law of the uniform distribution of primes over residue classes $\mod k$ ${\ displaystyle \ mod k}$ , insofar as

\lim _{s\to 1+}{\dfrac {\sum \limits _{p}{\dfrac {1}{p^{s}}}}{\ln {\dfrac {1}{s-1}}}}=1,

{\ displaystyle \ lim _ {s \ to 1 +} {\ dfrac {\ sum \ limits _ {p} {\ dfrac {1} {p ^ {s}}}} {\ ln {\ dfrac {1} { s-1}}}} = 1,}

if the summation is over all primes.

It is known that for any mutually prime numbers $l$ ${\ displaystyle l}$ and $k$ ${\ displaystyle k}$ row $\sum \limits _{p}{\frac {1}{p}}$ ${\ displaystyle \ sum \ limits _ {p} {\ frac {1} {p}}}$ where the summation is simple $p\equiv l{\pmod {k}}$ ${\ displaystyle p \ equiv l {\ pmod {k}}}$ , diverges.

Notes

↑ Yu. V. Linnik, A.O. Gelfand. Elementary methods in analytic number theory. - Fizmatgiz, 1962.

Literature

Postnikov M.M. Fermat's theorem. Introduction to the theory of algebraic numbers .. - M .: Nauka , 1986.

[1] Yu. V. Linnik, A.O. Gelfand. Elementary methods in analytic number theory. - Fizmatgiz, 1962.