I am currently interested in probability theory and analytic combinatorics. Here is a selection of my answers:

Limiting distribution of $\frac{1}{n} \sum_{k=1}^{n}|S_{k-1}|(X_k^2 - 1)$, where $X_k$ are i.i.d. standard normal

Does the sum $\sum_{n \geq 1} \frac{2^n\text{ mod } n}{n^2}$ converge?

Expected absolute difference between two i.i.d. variables

Does the sequence $x_{n+1} = -16 + 6x_n + \frac{12}{x_n}$ converge?

Proving $\lim_{x \to 0+} \sum_{n=0}^\infty (-1)^n/(n!)^x = \frac{1}{2}$

Solution of the functional equation $f(x) + f(y) = f(x + y + 2f(xy))$

How to prove the existence of this limit?