# A proof of a suprising limit on Dirichlet's theorem

Dirichlet's theorem states that there are infinitely primes exist, which are a residues class of $$r$$ of $$\pmod{n}$$ where $$n>2$$ or ($$p$$ is congruent to $$r \pmod{n}$$),and also there occurs unique numbers $$a$$ and $$b$$ which represents a prime number of the form $$a+nb$$ where $$n>0$$ and the numbers in these unique collection are relative prime to each other. Eeg there exist infinity many primes numbers of form $$6k+1$$,$$6k-1$$ such as $$5,7,11,13,19$$ and many more.

So my question; is there a formal elementary proof which shows that as $$x$$ tends to infinity, the proportion of number of primes less than $$x$$, which are congruent to $$r \pmod{n}$$, is to number of primes less than $$x$$ approaches to reciprocal of Euler totient function $$\varphi(n)$$?

• @ OP, if you want to add more context to the question, please consult this and this. Also, please check my edit ;) Commented Nov 1, 2020 at 12:57
• To make sure that I understand, the primary thing that you're asking for is an elementary proof, right? Commented Nov 2, 2020 at 18:13
• yeah ,any proof would do it. Commented Nov 2, 2020 at 20:51

Yes, there is an "elementary proof", namely the one given by Selberg in 1949 here on JSTOR. I would be wary for asking about elementary proofs, since most likely you are thinking that "elementary=simple" which is most definitely not true. For example, the elementary proof of the PNT given by Erdos and Selberg is one of the most complicated proofs I've seen, and the original "complex" one using the fact that $$\zeta(s)$$ is zero free for $$\Re(s)=1$$ is in my opinion much more simple and straightforward.