Analytic Number Theory

What is number theory? One might have thought that it was simply the study of numbers, but that is too broad a definition, since numbers are almost ubiquitous in mathematics. To see what distinguishes number theory from the rest of mathematics, let us look at the equation x + y = 15925, and consider whether it has any solutions. One answer is that it certainly does: indeed, the solution set forms a circle of radius √ 15925 in the plane. However, a number theorist is interested in integer solutions, and now it is much less obvious whether any such solutions exist. A useful first step in considering the above question is to notice that 15925 is a multiple of 25: in fact, it is 25 × 637. Furthermore, the number 637 can be decomposed further: it is 49 × 13. That is, 15 925 = 5 × 7 × 13. This information helps us a lot, because if we can find integers a and b such that a + b = 13, then we can multiply them by 5 × 7 = 35 and we will have a solution to the original equation. Now we notice that a = 2 and b = 3 works, since 2 + 3 = 13. Multiplying these numbers by 35, we obtain the solution 70 + 105 = 15925 to the original equation. As this simple example shows, it is often useful to decompose positive integers multiplicatively into components that cannot be broken down any further. These components are called prime numbers, and the fundamental theorem of arithmetic states that every positive integer can be written as a product of primes in exactly one way. That is, there is a oneto-one correspondence between positive integers and finite products of primes. In many situations we know what we need to know about a positive integer once we have decomposed it into its prime factors and understood those, just as we can understand a lot about molecules by studying the atoms of which they are composed. For example, it is known that the equation x+y = n has an integer solution if and only if every prime of the form 4m + 3 occurs an even number of times in the prime factorization of n. (This tells us, for instance, that there are no integer solutions to the equation x+y = 13475, since 13475 = 5×7×11, and 11 appears an odd number of times in this product.) Once one begins the process of determining which integers are primes and which are not, it is soon apparent that there are many primes. However, as one goes further and further, the primes seem to consist of a smaller and smaller proportion of the positive integers. They also seem to come in a somewhat irregular pattern, which raises the question of whether there is any formula that describes all of them. Failing that, can one perhaps describe a large class of them?We can also ask whether there are infinitely many primes? If there are, can we quickly determine how many there are up to a given point? Or at least give a good estimate for this number? Finally, when one has spent long enough looking for primes, one cannot help but ask whether there is a quick way of recognizing them. This last question is discussed in computational number theory; the rest motivate this article. Now that we have discussed what marks number theory out from the rest of mathematics, we are ready to make a further distinction: between algebraic and analytic number theory. The main difference is that in algebraic number theory (which is the main topic of algebraic numbers) one typically considers questions with answers that are given by exact formulas, whereas in analytic number theory, the topic of this article, one looks for good approximations. For the sort of quantity that one estimates in analytic number theory, one does not expect an exact formula to exist, except perhaps one of a rather artificial and unilluminating kind. One of the best examples of such a quantity is one we shall discuss in detail: the number of primes less than or equal to x. Since we are discussing approximations, we shall need terminology that allows us to give some idea of the quality of an approximation. Suppose, for example, that we have a rather erratic function f(x) but are able to show that, once x is large enough, f(x) is never bigger than 25x. This is useful because we understand the function g(x) = x quite well. In general, if we can find a constant c such that |f(x)| cg(x) for every x, then we write f(x) = O(g(x)). A typical usage occurs in the sentence “the average number of prime factors of an integer up to x is log log x+O(1)”; in other words, there exists some constant c > 0 such