Given the number of times we've mentioned Euler's work, you might think it is odd that the zeta function is named for Riemann. The next three chapters will try to counter that impression. In fact, what Riemann showed was that, despite Euler's quote above, there is a certain regularity to the sequence of prime numbers.

Georg Friedrich Bernhard Riemann (1826–1866). Riemann, just like Euler before him, began university studies as a student of theology, in accordance with his father's wishes, before switching to mathematics. Riemann had a very short career and produced only a few papers before he died of tuberculosis. In spite of that, he was one of the greatest mathematicians of all time.

For his Habilitationsvortrag, the advanced part of his Ph.D. degree, Riemann prepared three potential topics to lecture on. To his surprise, Gauss selected “On the Hypotheses that lie at the foundations of geometry.” The first part of the lecture describes the mathematical problem of defining n-dimensional space with an abstract concept of distance. In the second part, he discussed the question of the actual geometry of the universe we live in.

Traditional books on number theory that emphasize the algebraic aspects begin with the subject of modular arithmetic. This is a very powerful technique, based on the work of Fermat, Euler, and Gauss. The idea is to fix an integer, n, and group all the rest of the integers into one of n classes, depending on what remainder you get when you divide by n. A more elegant way of saying this is that two integers, a and b, are in the same class if n divides b − a or, in other words, b − a is an integer multiple of n. In this case, we write a ≡ b mod n. (This is pronounced “a is congruent to b modulo n.”) For example, with n = 6, 5 ≡ 17 mod 6, −2 ≡ 4 mod 6, and even 6 ≡ 0 mod 6. Because there are seven days in a week, the fact that 3 ≡ 10 mod 7 means that the 3rd of the month and the 10th of the month fall on the same weekday.

Example 14.0.1. The following steps show that ≡ is an equivalence relation.

1. Show that ≡ is reflexive, that is, that a ≡ a mod n.

2. Show that ≡ is symmetric; that is, if a ≡ b mod n, then b ≡ a mod n.

3. Show that ≡ is transitive; that is, if a ≡ b mod n and b ≡ c mod n, then a ≡ c mod n.

A Probability Argument

After this long detour through calculus, we are ready to return to number theory. The goal is to get some idea of how prime numbers are distributed among the integers. That is, if we pick a large integer N, what are the chances that N is a prime? A rigorous answer to this question is hard, so in this section we will only give a heuristic argument. The general idea of an argument based on probability is very old. Not only is it known not to be a proof (Hardy, Littlewood, 1922), but the way in which it fails to be a proof is interesting.

Because this will be an argument about probability, some explanation is necessary. If you flip a fair coin twelve times, you expect heads to come up about 6 = 12 × 1/2 times. You can think of this 6 as 1/2 + 1/2+ ⋯ +1/2, twelve additions. If you roll a fair die twelve times, you expect to roll a five about 2 = 12 × 1/6 times. The 2 is 1/6 added twelve times. This tells us what to do when the probability changes from one trial to the next. Imagine an experiment in which, at the kth trial, the chance of success is 1/k. If you repeat the experiment n times, how many successes do you expect? The answer is 1 + 1/2 + 1/3+ ⋯ +1/n = Hn. Because we already know that the Harmonic number, Hn, is about log(n) in size, we expect log(n) successes after n trials.

The techniques discussed in the previous chapters can be pushed a little further, at the cost of a lot of work. To make real progress, however, we need to study the prime numbers themselves. How are the primes distributed among the integers? Is there any pattern? This is a very deep question, which was alluded to at the beginning of Chapter 3. This Interlude makes a detour away from number theory to explain the ideas from calculus that we will need. It covers things I wish you had learned but, based on my experience, I expect you did not. I can't force you to read it, but if you skip it, please refer back to it later.

Linear Approximations

Although you might not notice, all of differential calculus is about a single idea: Complicated functions can often be approximated, on a small scale anyway, by straight lines. What good is such an approximation? Many textbooks will have a (rather unconvincing) application, something like “approximate the square root of 1.037.” In fact, almost everything that happens in calculus is an application of this idea.

For example, one learns that the graph of function y = f(x) increases at point x = a if the derivative f′(a) is positive. Why is this true? It's because of the linear approximation idea: The graph increases if the straight line approximating the graph increases. For a line, it's easy to see that it is increasing if the slope is positive. That slope is f′(a).