Primes in one reduced residue class have Dirichlet density 1 over phi(q)

Average the logarithms with the conjugate weights of the class $a$: $\frac{1}{\phi (q)}{\sum }_{\chi <mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>q}\overline{\chi (a)}\log L(s,\chi )={\sum }_{p,m\ge 1}(m{p}^{ms}{)}^{-1}\frac{1}{\phi (q)}{\sum }_{\chi <mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>q}\overline{\chi (a)}\chi (p{)}^m$. By [L1], the inner character sum is $1$ exactly when $p^m\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}q)$ and $0$ otherwise, so the left-hand side equals ${\sum }_{p^m\equiv a(q)}1\mathrm{/}(m{p}^{ms})$. The terms with $m\ge 2$ form a bounded tail as $s\downarrow 1$ because ${\sum }_{p,m\ge 2}1\mathrm{/}(m{p}^m)<\mathrm{\infty }$, and therefore $\frac{1}{\phi (q)}{\sum }_{\chi }\overline{\chi (a)}\log L(s,\chi )={\sum }_{p\equiv a(q)}{p}^{-s}+O(1)$.

For nonprincipal $\chi$, [L3] makes $\log L(s,\chi )=O(1)$ as real $s\downarrow 1$. For the principal character, the finite factor in [L3] tends to the positive value ${\prod }_{p\mid q}(1-p^{-1})$. The local fractional-part formula gives $\zeta (s)=1\mathrm{/}(s-1)+O(1)$, so $\log L(s,{\chi }_0)=\log (1\mathrm{/}(s-1))+O(1)$; this uses no global zeta continuation. Also $\overline{{\chi }_0(a)}=1$. Hence the average on the left side of step 1.1 is $\phi (q{)}^{-1}\log (1\mathrm{/}(s-1))+O(1)$. Comparing with step 1.1 yields the claimed asymptotic, which is exactly the Dirichlet density statement in Natural and Dirichlet density.