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$\zeta(s)$ denotes the Riemann zeta function. For $T>0$ large enough $$\int_{T}^{2T}|\zeta(\frac{1}{3} + it) \ \zeta(\frac{2}{3} + it)|dt = \ ? $$

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    $\begingroup$ See Ingham’s “Mean-Value Theorems in the Theory of the Riemann Zeta-Function” (Theorem A does the trick). In fact I believe your case reduces to work of Littlewood/Hardy-Littlewood on second moments by just using the functional equation + Stirling, but anyway. $\endgroup$
    – alpoge
    Commented Apr 28, 2019 at 1:15
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    $\begingroup$ What have you tried? $\endgroup$
    – Greg Martin
    Commented Apr 28, 2019 at 1:57

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