What’s Analytic Number Theory?

Analytic number theory is the application of the analysis of functions of a complex variable to problems involving number theory (the study of the integers). Now, when you first say that, it sounds completely deranged. After all, what can the properties of continuous functions of the complex numbers possibly have to do with the arithmetic of the whole numbers? Let me give a simple example that at least suggests that the idea of using analysis to study arithmetic might not be completely insane.

Here’s a question, then: how do we know there are infinitely many primes? Euclid’s standard proof says, suppose there were only finitely many primes . Think about the number . P must be factorable into primes, but it is obvious that none of the primes is a factor. Thus, our assumption that we had a complete list of primes must be wrong. The number of primes is therefore infinite.

So far, we haven’t used analysis at all, but now let’s look at a second proof of this theorem due to Euler. The function

defined at least for is called the Riemann zeta-function. If you think about the prime factorization of the denominators, the zeta-function can also be written as a product over all primes:

(The first equality comes from multiplying out all the denominators; the second is from the formula for a geometric series; the rest are just algebra.) Now think about the meaning of the sum/product identity

as from the right. The left hand side approaches the harmonic series

which diverges to infinity. (Not sure why? Check out the easy argument here.) The right hand side must therefore also approach infinity as . This can only happen if the product on the right hand side contains an infinite number of factors, so the number of primes must be infinite.

How does one react to this proof compared to Euclid’s? Although Euler’s approach may seem Byzantine in its complexity compared to Euclid’s gleaming lucidity, there are at least two reasons to pause in admiration at Euler’s method. In the first place, it makes a highly surprising connection between two seemingly utterly unrelated sides of human thought. In the second place, Euler’s method has buried in it more information than Euclid’s. Because the two sides of the sum/product identity (1) must approach infinity at the same rate, identity (1) not only tells us that there are infinitely many primes, but also places constraints on their density. That is, if primes were very common among the integers, then the right hand side of (1) would be quite large, while if primes were very sparse, then nearly all the factors in the product would be so close to 1 that the product would be fairly small. Analysing the sum on the left hand side of (1) therefore gives us some measure of how frequently primes occur among the integers.

Teasing this information out takes more refinement than I want to put into this introduction, but the eventual result of this line of reasoning is the stupidly named Prime Number Theorem of Hadamard and de la Vallée-Poussin, which states that for large n, the number of primes less than n is roughly . For a quick sense of the accuracy of this theorem, observe that the number of primes less than 1,000,000 is 78,498, while .

If you’d like an additional teaser showing some elementary properties of the zeta-function, you could follow that up here. The examples there may begin to convince you that an amazing amount of number theory is contained in the properties of this one function.

A final remark: the analytic methods sketched here can be vastly extended beyond what we’ve seen, providing a wealth of information on the distribution of primes as well as tantalizing partial results on the famous Goldbach Conjecture (that every even number is the sum of 2 primes). For many of these investigations, a central question turns out to be the location of the zeros of the zeta-function. It was conjectured by Riemann that the zeta-function has infinitely many zeros, that all the zeros have real part , and that the number of zeros with imaginary part between 0 and T is roughly

The first and last parts of this conjecture were proven by Hadamard and Mangoldt, resp. The middle part, that all zeros have real part , is called the Riemann Hypothesis, and is arguably the most famous and important unsolved problem in mathematics in the 20th century. (When David Hilbert was asked what his first question would be if he could awake like Barbarossa in 500 years, he said, “I would ask if the Riemann Hypothesis had been proved.”

Should you happen to prove it, by all means send me mail.

If you’d like to learn more about analytic number theory, three good places to start are Hardy and Wright’s Introduction to the Theory of Numbers, Tom Apostol’s Introduction to Analytic Number Theory, and Anatoly A. Karatsuba’s Complex Analysis in Number Theory. Try either amazon.com, or the library of our sister institution, Earlham College.