Motivation for using $L(1,\chi)$ in the proof of Dirichlet's Theorem Having read the proof of Dirichlet's Theorem on the infinitude of primes in arithmetic progressions, I am left wondering what his motivation for studying $L(1,\chi)$ was and why it is reasonable that studying L-functions could lead to a proof of the theorem. 
I have read and understood the proof given in Apostol's book on Analytic Number Theory, but still, I do not see the motivation for the proof. Does anyone have any knowledge on the topic and could provide some motivation for the proof?
 A: As you have seen, the main idea is to prove that $0<|L(\chi, 1)|< \infty$ when $\chi$ is a Dirichlet character. I'll explain why this is a natural thing to consider.
If $\chi$ is a Dirichlet character, then the (conditionally convergent) sum
$$L(1, \chi) = \sum_{n\geq 1} \chi(n) n^{-1}$$
can be written formally as the Euler product
$$L(1, \chi) = \prod_p (1-\chi(p)p^{-1})^{-1}.$$
In the world of infinite products, a product converges if its limit exists and is nonzero. Thus, the fact that $L(1, \chi) \neq 0, \infty$ can be re-stated by saying that this product converges, in the sense of convergence for infinite products. The fact that the product converges is then easily seen to be equivalent to the fact that the values $\chi(p)$ are roughly equidistributed along the unit circle. For example, in the case of the Riemann zeta function $\chi \equiv 1$, the values $\chi(p)=1$ all point in the same direction, and they "conspire" together resulting in $\zeta(1) = \infty$. For a nontrivial Dirichlet character, the fact that the product converges is therefore equivalent to the fact that the primes are roughly equidistributed modulo the conductor of $\chi$. This is why it is natural to look at $L(\chi, 1)$ - but don't let the sum fool you; the interesting thing is really the product!
