For $x \ge 1$, we define the expected value of the largest $k^{\text{th}}$-power divisor over integers $n \le x$ by $E_x(r^k) := \frac{1}{x} \sum_{n \le x} \max {{ r^k : r^k \mid n }}.$ Motivated by a question on MathOverflow concerning the square case ($k=2$) and a heuristic argument of Yuval Peres, we study the asymptotic behavior of $E_x(r^k)$ as $x \to \infty$. Peres’ heuristic predicts that $E_x(r^2)$ grows on the order of $\sqrt{x}$, but the associated error term is too large to determine the correct leading constant. We prove that $ E_x(r^2) = \frac{\zeta(3/2)}{3\zeta(3)} \sqrt{x} + O(\log x), $ where $\zeta(s) = \sum_{n=1}^{\infty} 1 / n^s$ is the Riemann zeta function. This result identifies the correct leading constant supported by numerical evidence. More generally, for any $k \ge 2$, we show that $ E_x(r^k) = \frac{\zeta\left(\frac{k+1}{k}\right)}{(k+1)\zeta(k+1)}\, x^{1/k} + O(\log x). $ The constants arise naturally from zeta functions associated with $k$-free integers and quantify the average size of the largest $k^{\text{th}}$-power divisor.

https://www.overleaf.com/read/qbwjysmbpzbb#345bb6 https://mathoverflow.net/questions/379550/size-of-largest-square-divisor-of-a-random-integer