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

Is there any $y>0$ such that $F_{y}(x) \rightarrow 0$ as $x\rightarrow \infty$ ?

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 $x\rightarrow \infty$ ?

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

Is there any $y>0$ such that $F_{y}(x) \rightarrow 0$ as $x\rightarrow \infty$ ?

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

Is there any $y>0$ such that $F_{y}(x) \rightarrow 0$ as $x\rightarrow \infty$ ?