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In http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf on page 6 (top) the author states that: $$ \sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} = \frac{\log \log x}{\phi(l)} + O \left ( \frac{\log l}{l} \right ) $$ He refers to http://www.math.dartmouth.edu/~carlp/Amicable1.pdf for an example, but here it is only proved that: $$ \sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} = \frac{\log \log x}{\phi(l)} + O \left ( \frac{\log l}{\phi(l)} \right ) $$ However, this estimate is not enough to finish the proof in the original article. Does anyone know how to prove the first statement?

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Correction: The second estimate is enough to finish the proof in the original article. But I'll still like to hear if anyone can prove the first estimate on the error-term. –  Mathias Bæk Tejs Knudsen Oct 25 '12 at 19:37

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