# Evaluation of a number-theoretical sum

1. Aug 9, 2005

### veloz

I need some help evaluating the following sum.

Let $$M$$ be some large integer

$$Q = \prod_{\substack{p \le \frac{M}{3} \\ (p,M)=1}}{p}$$

$$\mu(n)$$ be the Möbius function

and

$$r(n)=\left\{\begin{array}{cc}1,&\mbox{ if } n \ is \ prime\\0, & \mbox{ } otherwise\end{array}\right.$$

Then I am interested in the sum

$$\sum_{\substack{d|Q \\ d < M}}{\frac{\mu(d)}{d}\sum_{\substack{e|\frac{Q}{d} \\ e(1+2r(e)) < M - d (1 + 2r(d)) }}{\frac{\mu(e)}{e}}}$$

The related sum
$$\sum_{\substack{e|Q \\ e < x}}{\frac{\mu(e)}{e}}}$$

is evaluated in www3.sympatico.ca/robert.juricevic/sieve.ps
as lemma 3.12 on page 25.

Any help would be greatly appreciated