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If $a$ and $q$ are given coprime positive integers, what is the best known error term for $$ \sum_{p<x,\,p\,\text{is prime},\,p\equiv a \pmod q} \frac{\log p}p-\frac{\log x}{\varphi(q)}? $$ Is it, say, proven to be $O(1)$?

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    $\begingroup$ Yes it is $O(1)$. You can see this by using the Brun-Titchmarsh theorem (and partial summation) to bound the sum over primes in the range $x\le \exp(\phi(q))$, and then using the prime number theorem in arithmetic progressions and partial summation for all larger $x$. $\endgroup$
    – Lucia
    Oct 27, 2013 at 13:08
  • $\begingroup$ One more remark: it is hard to avoid the error term $O(1)$ since it may happen for example that $a$ is itself a small prime. If you omit this first term, that is sum over $q<p\le x$ then one should be able to get better error terms. $\endgroup$
    – Lucia
    Oct 27, 2013 at 13:14
  • $\begingroup$ Well, in my situation $q$ is given, so I wonder only about large values of $x$. But indeed, it just follows from any reasonable remainder estimate in PNT for arithmetic progressions. $\endgroup$ Oct 27, 2013 at 13:20

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