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I've seen that on the line $Re(s)=\frac{1}{2}$, $\zeta(s)=O(t^{\frac{1}{4}})$ where, as usual, $s=\sigma+it$. My teacher has told me that this can be derived directly from the functional equation of the zeta function, i.e., $\zeta(s)=2^s\pi^{s-1}\sin(\frac{\pi s}{2})\Gamma(1-s)\zeta(1-s)$. Can someone please tell me how that estimate is obtained from this equation? If you can give a link to any pdf where this has been proved in detail, that'd also be fine. Thanks in advance.

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If we write the functional equation in the usually compact way $$\zeta\left(s\right)=\chi\left(s\right)\zeta\left(1-s\right) $$ holds the following theorem known as approximate functional equation:

Theorem: If $h$ is a positive constant, if $x,y$ are two real numbers such that $x>h>0,\, y>h>0$ and if $s=\sigma+it$ with $0<\sigma<1$ and $t=2\pi xy$ holds $$\zeta\left(s\right)=\sum_{n\leq x}\frac{1}{n^{s}}+\chi\left(s\right)\sum_{n\leq y}\frac{1}{n^{1-s}}+O\left(x^{-\sigma}\log\left|t\right|\right)+O\left(\left|t\right|^{1/2-\sigma}y^{\sigma-1}\right).$$ and now your claim follows if we take $\sigma=1/2$, $x=y=\sqrt{t/\left(2\pi\right)}$.

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