Dirichlet series of the reciprocal radical function - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:22:05Z http://mathoverflow.net/feeds/question/73193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73193/dirichlet-series-of-the-reciprocal-radical-function Dirichlet series of the reciprocal radical function unknown (google) 2011-08-18T23:57:41Z 2011-08-19T23:28:44Z <p>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 $A>0$ ?</p> http://mathoverflow.net/questions/73193/dirichlet-series-of-the-reciprocal-radical-function/73195#73195 Answer by Gjergji Zaimi for Dirichlet series of the reciprocal radical function Gjergji Zaimi 2011-08-19T00:34:55Z 2011-08-19T23:28:44Z <p>This problem was studied by De Bruijn, see</p> <blockquote> <p>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</p> </blockquote> <p>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 <a href="http://arxiv.org/abs/math.NT/0605019" rel="nofollow">"Idempotents and Nilpotents Modulo n"</a> for a discussion of this and similar problems (and more complete references).</p>