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It is well known that the Lindel\"of Hypothesis is equivalent to the statement that $$\frac 1T\int_0^T|\zeta(1/2=it)|^{2k} =O(T^\epsilon)$$ for all positive integers $k$ and all positive real $\epsilon$, and these fact are known for $k=1, 2$. May I ask you for references to the proofs.

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  • $\begingroup$ In papers and chapters (Titchmarsh and others) about the Dirichlet divisor problem $\endgroup$ – reuns Dec 6 '18 at 20:16
  • $\begingroup$ Do you mean the equivalence? I am interested in the proofs of the estimate for $k=1, 2$. $\endgroup$ – Durac Dec 6 '18 at 20:27
  • $\begingroup$ zulfahmed.files.wordpress.com/2018/08/… p.74-75 $\endgroup$ – reuns Dec 6 '18 at 20:40
  • $\begingroup$ Thanks! Just a remark: this is the chapter of Titchmarsh titled 'Mean value theorems'. (You have answered my question, I am ready to accept.) $\endgroup$ – Durac Dec 6 '18 at 20:51

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