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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 de-
termine 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 the 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. Well, there are answers and answers. To those
of us who have witnessed the use of generating functions this is a kind
of answer, but to those of us who haven’t, this is simply a restatement
of the question. Perhaps the best way to understand the use of the
analytic method, or the use of generating functions, is to see it in
action in a number of pertinent examples. So let us take a look at
some of these.

Addition Problems

Questions about addition lend themselves very naturally to the use of
generating functions. The link is the simple observation that adding
m
n
m and n is isomorphic to multiplying z and z . Thereby questions
about the addition of integers are transformed into questions about
the multiplication of polynomials or power series. For example, La-
grange’s beautiful theorem that every positive integer is the sum of

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