# Size of largest square divisor of a random integer

Let $$x$$ be an integer picked uniformly at random from $$1 \ldots N$$. Write $$x = r^2 t$$ where $$t$$ is square-free. How does the expected value of $$r$$ scale with $$N$$? Is anything known about the variance of $$r$$?

The asymptotic frequency of square-free integers is known to be $$6/\pi^2$$, see [1].

Denote by $$P_n$$ the uniform distribution on $$[1,n]$$ and by $$E_n$$ the corresponding expectation. Then $$E_n(r)=\sum_{k \le \sqrt{n}} k P_n(r=k) \sim \sum_{k \le \sqrt{n}} k \cdot \frac{1}{k^2} \cdot\frac{6}{\pi^2} \sim \frac{3}{\pi^2} \log(n) \,,$$ where $$A \sim B$$ means that $$A/B \to 1$$ as $$n \to \infty$$. (In particular for $$n=10^{10}$$ the mean $$E_n(r)$$ is close to 7.) Also, $$E_n(r^2)=\sum_{k \le \sqrt{n}} k^2 P_n(r=k) \sim \sum_{k \le \sqrt{n}} \frac{6}{\pi^2} \sim \frac{6\sqrt{n}}{\pi^2} \,,$$ so the variance of $$r$$ is asymptotic to $$6\sqrt{n}/\pi^2$$ as well.

• I have never seen $\log_e 10 \approx \frac{7}{30}\pi^2$ being useful before: both about $2.303$ Dec 23, 2020 at 19:02

Just a little empirical data up to $$N=10^{10}$$, three superimposed random runs. Growing roughly linearly w.r.t. $$\log_{10} N$$ within that range.

• What is the slope and how close to $6/\pi^2$ is it?
– smci
Dec 23, 2020 at 10:36
• @smci: Slope is about $0.71$, but I plotted w.r.t. $\log_{10}$. The constant in Yuval's $E_n(r)$ is $3/\pi^2$, and $(3/\pi^2) \log(10) \approx 0.70$; so that matches. And, as Yuval says, for $N=10^{10}$, the expected value of $r$ is about $7$, which accords with my chart, and with $0.70 \times 10 = 7$. Dec 23, 2020 at 14:15