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To prove that $$ \frac{\#(\text{primes}~~ qn+a \leq x)}{\pi(x)/\phi(q)}\to 1~ \text{as} ~ x\to \infty, $$ that there are roughly the same number of primes in each residue class $a$ if $(a,q)=1$ without some version of Dirichlet characters/series might be hard. There is no special virtue of using $Li(x)$ here, any function $\sim \pi(x)$ will work. ...


I apologize for the delay in writing this up. Here is a proof of the remaining inequality (the one not proved in your answer, ireallydonknow). As I mentioned in a comment, this proof is a simple application of ideas found on page 7 of Hardy & M. Riesz's The General Theory of Dirichlet's Series. Here are a few remarks (mostly paraphrasing that book) to ...

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