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We have $\frac{1}{2} < \sigma < 1$ and
$$ f(n) = \Bigl|\, 1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr| $$ . My goal is proving this statement that $|f(n)|$ is $O(\log n)$ but I'm not certain that this is true. some bounds like $O(n^{1-\sigma})$ are simple to prove but my desired bound is $\log n$.

Any help?

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    $\begingroup$ This is equivalent asking for the bound $\zeta(\sigma+it) = O( \log t )$ for integer values of $t$, which is basically a variant of the Lindelof hypothesis. $\endgroup$
    – Terry Tao
    Commented Aug 26, 2015 at 18:12
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    $\begingroup$ Actually, from looking at Montgomery's paper ams.org/mathscinet-getitem?mr=460255 it seems that the bound $\zeta(\sigma+it) = O(\log t)$ cannot hold for all sufficiently large $t$; presumably the argument can extend to the case when $t$ is restricted to be integer. $\endgroup$
    – Terry Tao
    Commented Aug 26, 2015 at 18:18
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    $\begingroup$ However, mean value theorems will show that the desired bound is true for "most" values of n. See e.g. Titchmarsh's book on the zeta function. $\endgroup$
    – Terry Tao
    Commented Aug 26, 2015 at 18:20
  • $\begingroup$ Thanks @TerryTao. I have a question: Have you noticed that I want to have the logarithmic order just on a fixed vertical line which is in the right of the critical line $Re(s) = 0.5$, not on the whole of the critical strip or on the critical line? (According to that which the Lindelof Hypothesis is for the growth of zeta on the critical line) $\endgroup$
    – Mohammad Ghiasi
    Commented Aug 27, 2015 at 14:24
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    $\begingroup$ The growth of zeta (after deleting the pole at s=1) on the critical line is related to the growth on other lines by Lindelof's theorem: en.wikipedia.org/wiki/Lindel%C3%B6f%27s_theorem . Note also that Montgomery's paper cited above also applies to lines to the right of the critical line. $\endgroup$
    – Terry Tao
    Commented Aug 27, 2015 at 18:07

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