# Reference requested for $\lim_{n \rightarrow \infty} \frac{\sum_{i=1}^{n} \bar{s}(i)}{n^2} = \frac{\pi^2}{30}$

While analysing the average runtime of an algorithm, I came across the following identity, and would like to know if anybody knows of any references for it?

For $i \in \mathbb{N}$, let $\bar{s}(i)$ denote the square-free part of $i$, eg., $\bar{s}(12) = 3$ (and $\bar{s}(1)=1$). Then $$\lim_{n \rightarrow \infty} \frac{\sum_{i=1}^{n} \bar{s}(i)}{n^2} = \frac{\pi^2}{30}.$$

Many thanks.

• You have $\prod_p (1+p/p^s+1/p^{2s}+p/p^{3s}+\cdots)$, alternately $1$ and $p$ in coefficients, and likely can write this as a $\zeta$ expression. Jun 8 '11 at 21:36
• I guess it is $\zeta(2s)\zeta(s-1)/\zeta(2s-2)$. Jun 8 '11 at 21:46

If my understanding is correct, for "squarefree part" can be "squarefree kernel" in other cases, the generating Dirichlet series is $${\zeta(2s)\zeta(s-1)\over\zeta(2s-2)}=\prod_p\biggl(1+{p\over p^s}+{1\over p^{2s}}+{p\over p^{3s}}+\cdots\biggr)=\sum_n{\bar s(n)\over n^s}$$ alternating $1$ and $p$ as the coefficients, which is a $\zeta$ quotient as indicated. The residue at $s=2$ is $\zeta(4)/\zeta(2)={\pi^4/90\over\pi^2/6}={\pi^2\over 15}$, so that by Perron's formula $$\sum_{n\le X} \bar s(n)={1\over 2\pi i}\int_{(\sigma)}{\zeta(2s)\zeta(s-1)\over\zeta(2s-2)}{X^s ds\over s} \sim {\zeta(4)\over 2\zeta(2)}X^2={\pi^4/90\over2\pi^2/6}X^2={\pi^2\over 30}X^2,$$ with usual conditions about convergence in vertical strips, which are OK here. Dividing by $X^2$ gives the desired limit.
• Junkie, by Perron's formula we have $$\sum_{n\le X} \bar s(n)=\int_{(3)}\frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)}\frac{X^s}{s}\,ds\sim \frac{\pi^2}{30}X^2,$$ which is the limit conjectured by the OP. So there is no need to do any partial summation: the extra factor $2$ comes from the $s$ in the denominator of the integrand. Jun 8 '11 at 22:41