Suppose that $t > 1$ is fixed. I'm trying to find a sequence $\{x_k\}_{k=1}^\infty$ possibly dependent on $t$ that satisfies the following conditions:
1) $x_k \to 0$ as $k\to \infty$;
2) $\displaystyle\sum_{k=1}^\infty \dfrac{\sqrt[t]{x_k}}{k} < \infty$;
3) $t^{-\tfrac{n}{2}} \leq C_t \; x_1x_2 \cdots x_n,$ for every $n \geq 1$ and some constant $C_t$.
If we put $x_k = \frac{1}{k}$, then the first two conditions are fulfilled, but I can't get the third fails since we get a factorial on the right hand side which falls off much more quickly then the power on the left hand side.
If we put $x_k = t^{-k}$, the first two conditions are again fulfilled, and in the third we get $ t^{-\frac{n(n+1)}{2}}$ on the right hand side. I don't see a way to find a $C_t$ such that the estimate holds.
If we put $x_k = \frac{1}{\log(1+k)}$, then the sum in $2)$ diverges.
Any better ideas?
What about $$t^{-tE[\sqrt k]}$$