Applications of generating functions to number theory

Question

I am familiar (at least at a cursory level) with the extensive role generating functions play in the theory of partitions. What are some other prominent applications of generating functions to number theory?

Answer 1

This answer is by no means exhaustive, it's presumably hardly more than a startup for your question.

Considering the theory of partitions as part of additive number theory which uses quite often analytical methods, we can say that generating functions are a ubiquitous tool in analytic number theory and not only within the theory of partitions. To underpin this statement I like to quote from D.J.Newman's Analytic Number Theory.

From D.J.Newman's Analytic Number Theory, chapter I: The Idea of Analytic Number Theory

The most intriguing thing about Analytic Number Theory (the use of Analysis, or function theory, in number theory) is its very existence! How could one use properties of continuous valued functions to determine properties of those most discrete items, the integers. Analytic functions? What has differentiability got to do with counting? The astonishment mounts further when we learn that complex zeros of a certain analytic function are the basic tools in the investigation of the primes. The answer to all this bewilderment is given by the two words generating functions.

The following chapter II deals with partition functions, but then follow (besides other ones) chapters about the Waring Problem, about $L$-Series and the prime number theorem.

In multiplicative number theory generating functions take the form of Dirichlet Series to answer questions using sieve methods, questions about arithmetic functions or the distribution of primes to name just a few areas.

In my job I've sometimes to do with generating functions in cryptography which could together with the usage of generating functions in computational number theory also be seen as part of number theory.

Answer 2

One can view $q$-expansions of modular forms as a kind of generating function. They contain information on many number theoretic objects such as partitions, divisor sums, point counts on elliptic curves, representation numbers of quadratic forms, class numbers etc.