Dirichlet theorem

Question

Can anyone give a simple number theory proof for the Dirichlet theorem? Statement of Dirichlet theorem:given any two numbers a and b whose g.c.d is 1,Prove that infinitely many primes exist in the series a,a+b,a+2b,.....

Answer 1

There is no "simple number theory" proof anywhere and it would be nice if some "elementary" proof was presented.
However there is an elementary proof here http://www.jstor.org/stable/1969454?seq=1#page_scan_tab_contents but is extremely complicated....

Answer 2

the proof is not complicated once your understood the L-series for Dirichlet characters and the discrete Fourier transform.

for a prime number $q$ let $n \equiv g^{l(n,g,q)} \pmod q$ where $g$ is a generator of $(\mathbb{Z}/q\mathbb{Z},\times)$ and $l(n,g,q)$ is the discrete logarithm of $n$ in base $g$ modulo $q$. then for any $m \in \mathbb{N}, \omega(m)= e^{2 i \pi m /(q-1)}$:

$$F(s,g,q,m) = \sum_{n=1}^\infty \omega(m)^{l(n,g,q)} n^{-s} = \prod_{p \in \mathcal{P}} \frac{1}{1-\omega(m)^{l(p,g,q)} p^{-s}}$$

$$\log F(s,g,q,m) = \sum_{p^r} \frac{\omega(m)^{l(rp,g,q)} p^{-sr}}{r} = \mathcal{O}(1) + \sum_{p} \omega(m)^{l(rp,g,q)} p^{-s} \qquad(\Re(s) > 1/2)$$ where $p$ and $p^r$ run over primes and power of primes respectively.

because of the definition of $\omega(m)$ and because the discrete logarithm is a permutation $\phi$ of $\mathbb{Z}/(q-1)\mathbb{Z}$, we get that for any sequence $c(m)$ on $\{0\ldots q-2\}$ : $ \ \sum_{m=0}^{q-2} c(m) \omega^{l(m,g,q)}$ is the discrete Fourier transform of $c(\phi^{-1}(m)$, i.e. that for any $q-1$ periodic function on the integers there exists some coefficients $H$ such that $h(n) = \sum_{m=0}^{q-2} H(m) \omega^{l(n,g,q)}$, in particular let $D(m)$ such that $\delta_{n \equiv a \pmod q} = \sum_{m=0}^{q-2} D(m) \omega^{l(n,g,q)}$ :

$$\sum_{p, \ p \equiv a \pmod q} p^{-s} = \mathcal{O}(1) + \sum_{m=0}^{q-2} D(m) \log F(s,g,q,m) \qquad(\Re(s) > 1/2)$$

and the result follows from the fact that $\log F(s,g,q,m)$ is holomorphic at $s=1$ except for $m=0$, and that $D(0) \ne 0$ (because of the property of the discrete Fourier transform of a Dirac), i.e $\sum_{p, \ p \equiv a \pmod q} p^{-s}$ has a pole at $s=1$, so there are intinitly many primes $\equiv a \pmod{q}$.