For two large (10^6) integers A and B, find the number of set bits in the result of their multiplication
I would assume that a very large product is at least say up to 128 bits (product of two 64 bit integers). In that case there are simple algorithms that calculate the higher and the lower 64 bits of the result, and then you count individually how many bits are set in the higher 64 bits, and how many in the lower 64 bits, and add the two numbers.
If your very large product is the product of say two numbers of $10^6$ bits each, these numbers could each be split into say 15,625 64 bit integers, and the product can be calculated easily with O(n^2) operations for n = 15,625. Or it can be calculated faster but with much more difficulty with a more complicated algorithm. This gives a result of 31,250 64 bit integers, and you can easily count the number of bits set in each integer, and add those numbers.