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Let $\chi$ be a primitive Dirichlet character of conductor $q>1$. One may use partial summation to prove an upper bound of the form (I hope I am right) $$ \sum_{n\leq X} \chi(n)n^{-1/2-it} \ll \sqrt q \min (\left|t\right|, \sqrt X)\log q. $$

What is the best known bound for the above sum?

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  • $\begingroup$ It depends on what you want to do ? With this elementary method you'll get $\mathcal{O}(t^{1/2+\epsilon})$. Using convexity you'll get $\mathcal{O}(t^{1/4+\epsilon})$ $\endgroup$
    – reuns
    Commented Apr 14, 2017 at 22:34
  • $\begingroup$ Thanks. What I want is a nontrivial upper bound in terms of the three parameters $q,X,t$. $\endgroup$
    – U Ser
    Commented Apr 15, 2017 at 3:58
  • $\begingroup$ You can adapt it to partial sums. What do you want to do with those bounds ? $\endgroup$
    – reuns
    Commented Apr 15, 2017 at 4:00
  • $\begingroup$ Ok thanks. I will try to adapt it. It occurs in another sum I want to estimate, and if I have a good bound for this, I hope I can have a nontrivial bound for the "other" sum. It is a bit technical I cannot say it in detail. The above trivial bound happens to be a bit too weak for my purpose. $\endgroup$
    – U Ser
    Commented Apr 15, 2017 at 4:06
  • $\begingroup$ Really it is hard to help if you don't say the bound you need.. There are a lot of non-trivial bounds for those partial sums in the critical strip $\endgroup$
    – reuns
    Commented Apr 15, 2017 at 4:24

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