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Denote by $\zeta$ the Riemann zeta function. Define $$F_{y}(x)= \int_{-\infty}^{\infty} \frac{x^{iu}\zeta(1/2 + it + iu)}{u^2 + y^2} \mathrm{d}u.$$

For some fixed real number $t$, is there any $y>0$ such that $F_{y}(x) \rightarrow 0$ as $x\rightarrow \infty$ ?

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  • $\begingroup$ What is $t$ (or $τ$ in the title)? $\endgroup$
    – LeechLattice
    Commented Jul 1, 2019 at 1:16
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    $\begingroup$ @Bullet51, $\tau$ in the title is simply a dummy real variable, and $t$ is some fixed real number. $\endgroup$
    – user140392
    Commented Jul 1, 2019 at 1:28

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