I’ll give a birds-eye-view of how the non-vanishing a (generating) function translates to other pieces of information.  We’ll start with how a zero-free region of the Riemann zeta function implies the prime number theorem and how the Riemann hypothesis is equivalent to a smaller error in the prime number theorem.   We’ll then move on to further examples in statistical mechanics and probability.  No knowledge of anything is assumed, and it’s likely I won’t get through everything in the title.