By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$.
I would like to have it clarified what are the analogous equivalent statements for primes in arithmetic progressions. In particular, I would love to have specific references to papers or books in which the following equivalences are proved (or at least stated explicitly):
- Dirichlet's theorem, that there are infinitely many primes in any admissible arithmetic progression (mod $q$). is equivalent to the nonvanishing of $L(1,\chi)$ for all Dirichlet characters $\chi\pmod q$. [Edit: as has been pointed out, this nonvanishing is more likely equivalent to the equality of the Dirichlet densities of primes among the reduced residue classes (mod $q$). What analytic statement could be equivalent to the mere infinitude of primes in all such classes?]
- The prime number theorem in arithmetic progressions $\pi(x;q,a)\sim x/(\phi(q)\log x)$ is equivalent to the statement that $L(s,\chi)$ does not vanish on the line $\Re s=1$ for all Dirichlet characters $\chi\pmod q$.
Of course, if these statements themselves are incorrect, I would like to be corrected as well as being pointed to the literature.