3 problems

Let $\Omega(n)$ denote the total number of prime factors of $n$, counted with multiplicity, let $[N]=\{1,\ldots,N\}$, and for a finite set $S$ write…

Let $\mathbb{P}_\ell$ denote the set of integers with the relevant almost-prime parameter $\ell$, let $\1_{\mathbb{P}_\ell}$ be its indicator, and let $\overline{\pi}_\ell(N)$ deno…

Let $X$ be a positive random variable, and write … where $\lfloor\log_2 X\rfloor$ and $\{\log_2 X\}$ are respectively the whole and fractional parts of $\log_2 X$. Let $A$ and…