Let $c_q(n)$ be the Ramanujan sum, and let $\tau(n)$ be the divisor function. Is there an asymptotic formula for $$\sum_{n\le x}\tau(n)c_q(n)$$ with error terms that do not depend on $q$?
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1$\begingroup$ Substituting $c_q(n)=\sum_{d\mid (q,n)}d\mu(q/d)$ one can reduce this problem to the divisor problem in arithmetic progressions. This problem is well-studied, see Sándor, Jó.; Mitrinović, D. S. & Crstici, B. Handbook of number theory. I, § II.16 The divisor problem in arithmetic progressions. $\endgroup$– Alexey UstinovCommented Jan 3, 2017 at 9:28
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$\begingroup$ Do you know of any other references for that problem? $\endgroup$– Mayank PandeyCommented Jan 3, 2017 at 21:12
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$\begingroup$ This "handbook" contains different references. What is wrong with them? $\endgroup$– Alexey UstinovCommented Jan 3, 2017 at 23:51
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$\begingroup$ I don't have access to it. $\endgroup$– Mayank PandeyCommented Jan 4, 2017 at 3:10
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$\begingroup$ gen.lib.rus.ec booksee.org $\endgroup$– Alexey UstinovCommented Jan 4, 2017 at 4:29
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