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I recently saw on p.$458$ of Montgomery-Vaugahn's ''Multiplicative number theory'' that the inequality

$$\int_{1}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2})$$

holds uniformly for $T\geq 2$, where $\zeta$ denotes the Riemann zeta function. From this result, it clearly follows that

$$\Bigg| \int_{1}^{T} \zeta(1/2+it) \mathrm{d}T \Bigg|\geq T^{1/2}$$

for all $T\geq T_0$. I need help on finding the $T_0$.

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  • $\begingroup$ And what is your question? $\endgroup$
    – András Bátkai
    Commented Oct 27, 2018 at 14:56
  • $\begingroup$ @AndrásBátkai, my question (put otherwise) is does anyone know the $T_0$ that i'm looking for. I thought it was quite clear. $\endgroup$
    – user123305
    Commented Oct 27, 2018 at 16:15
  • $\begingroup$ I'm voting to close this question because the person behind all these user accounts does not seem to be interested in returning to this question, and has not been recently active under the user account used to post this question $\endgroup$
    – Yemon Choi
    Commented Dec 22, 2018 at 4:58
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    $\begingroup$ I'm voting to close this question again, since the person behind this and other accounts seems unwilling to admit to sockpuppeting $\endgroup$
    – Yemon Choi
    Commented Sep 11, 2019 at 0:09

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