Highly composite numbers are those positive integers with more divisors than any smaller number. In other words, they are the record setting numbers for the divisor function. The sequence begins 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... .

Now, the divisor function τ(n) counts the number of divisors of n, including 1 and itself. If n has a prime factorization that is the product of p_{i}^{ai}, with i running from 1 to k, then τ(n)=(a_{1}+1)(a_{2}+1)...(a_{k}+1).

We can generalize this function by replacing 1 in this product by a complex z.

We could say τ_{z}(n) = (a_{1}+z)(a_{2}+z)...(a_{k}+z).

Then we can define highly z-composite numbers to be those positive integers for which |τ_{z}(n) | is greater than |τ_{z}(j)| for all j < n.

Also, for each n, we can consider the set z of complex numbers for which n is highly z-composite. For most n, there will be no such z. But for some, this set might be quite interesting.

I will have to draw some pictures.