Problem Statement

The Mertens function M(x) is the count of square-free integers up to x that have an even number of prime factors, minus the count of those that have an odd number. It is an extension of the Möbius function. Given the Möbius function μ(n), the Mertens function M(x) is the sum of the Möbius numbers from n == 1 through n == x.

This is not code golf.   The stackexchange link is provided as an algorithm reference, not as a guide.

Let's start with the solution:

mu    =: 0:`(1 - 2 * 2|#@{.)@.(1: = */@{:)@(2&p:)"0
M     =: +/@([: mu 1:+i.)

m1000 =: (M"0) 1+i.1000
zero  =: +/ m1000 = 0
cross =: +/ (-.*.1:|]) m1000 ~: 0 

echo 'The first 99 Merten numbers are'
echo 10 10$ __, 99{.m1000
echo 'M(N) is zero ',(":zero),' times.'
echo 'M(N) crosses zero ',(":cross),' times.'
exit''