Define $ rad(n):=\prod_{p|n}p $ and $a_n:=\frac{n}{rad(n)}.$ For example $a_n=1$ whenever n is a squarefree integer. The assosiated Dirichlet series $$F(s):=\sum_{n} \frac{a_n}{n^s}=\prod_{p} (1+\frac{1}{p^s} \frac{1}{1-p^{1-s}})$$ has abcissa of convergence $s_0=1.$ Are there any results regarding the distribution of $a_n$, e.g. whether $\sum \limits_{n \leq x} a_n \ll x (\log x)^A $ for some real constant $A>0$ ?


This problem was studied by De Bruijn, see

N. G. de Bruijn and J. H. van Lint, "On the number of integers $\le x$ whose prime factors divide $n$", Acta Arith. 8 (1963) 349–356

The main result is that $$\log\left(S(x)\right)=\log\left(\sum_{n\le x} \frac{1}{rad(n)}\right)\sim \left(\frac{8\log x}{\log \log x}\right)^{1/2}$$ and that $$\sum_{n\le x} \frac{n}{rad(n)}=o\left(xS(x)\right).$$ See also the article "Idempotents and Nilpotents Modulo n" for a discussion of this and similar problems (and more complete references).

  • $\begingroup$ I take it the appearances of $n$ in the last term of the first display should instead be $x$? $\endgroup$ – Gerry Myerson Aug 19 '11 at 12:52

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