I will discuss a problem I encountered over a decade
ago, and worked on via internet with someone I (alas) never met.
It involves a mix of number theory, real analysis, hard-core
computation, and some slightly perplexing results.
In brief, we begin with a function expressed as a certain infinite
product; Arnold Knopfmacher encountered it in an attempt to
approximate the number of irreducible factors of univariate
polynomials over Galois fields and raised the querstion of how to
obtain a certain limit to this function. We derive and execute an
effective algorithm for the task at hand. We'll also indicate why
the most "obvious" approach does not work well in practice, or at
all in theory. |