In spite of the important relation between ζ(s) and the sequence of prime numbers, Riemann was not especially interested in number theory. His 8-page paper of 1859, entitled *On the Number of Primes Less Than a Given Magnitude* was, in fact, the only paper he published on number theory – but what a paper! Its actual focus was Riemann’s more abiding interest, the theory of complex functions. Yet in dealing with the function theory of ζ(s), he not only established some of its key properties, laying essential groundwork for the eventual proof almost 40 years later of the prime number theorem, but he also stated, quite off-handedly, his hypothesis which was to become perhaps the most notorious unsolved problem in mathematics for the better part of the last century. There is probably no other short paper in mathematics that has stimulated as much fruitful work as this one of Riemann’s (excepting perhaps the mere marginal comment which became known as Fermat’s Last Theorem).

To begin with, Riemann showed that ζ(s), as defined by either the Dirichlet series or the Euler product only in case Re(s) > 1, actually had an “analytic continuation” to the entire complex plane except for a single singularity at s=1. (That singularity was to be expected, given that the Dirichlet series reduces to the Taylor expansion of -log(1-s) at s=1.) Riemann did not use an abstract argument based on extending the function by means of defining it piecemeal on overlapping disks. (This procedure originated in the work of Karl Weierstrass (1815-97), which came a little later.) Instead, Riemann came up with an explicit formula for ζ(s) which actually made sense, and defined ζ(s), for all s ≠ 1. That formula can be written as:

ζ(s) = -Γ(1-s)(2πi)

^{-1}∫_{C}(-x)^{s-1}/(e^{x}-1) dx

This is what’s known as a “contour integral”, which is an integral that is evaluated on a path in the complex plane chosen to avoid the singularity of the integrand. In this formula, Γ(s) is Euler’s gamma function, which is defined by

Γ(s) = ∫

_{0≤x<∞}e^{-x}x^{s-1}dx

Although Γ(s) has a definition in the form of a definite integral, it’s more intuitively thought of as a generalization of the factorial function, since Γ(n+1) = n! for integers n ≥ 0, and in fact Γ(s+1) = (s+1)Γ(s) for all s.

Even more importantly, Riemann proved that ζ(s) satisfies a “functional equation” which relates its values at s to its values at 1 – s. The functional equation is this:

ζ(1-s) = (π

^{1/2-s}Γ(s/2) / Γ((1-s)/2)) ζ(s)

Since Γ(s) appears in Riemann’s integral formula for ζ(s), it’s not surprising that it also appears in the functional equation. The equation isn’t too hard to prove, given known properties of Γ(s) and standard facts of complex function theory. In fact, Riemann provided two proofs of the equation in his paper. There are many other proofs. E. C. Titchmarsh in his *Theory of the Riemann Zeta Function* gave seven proofs.

Various important facts about ζ(s) follow immediately from the functional equation. Clearly, Γ(s) plays a significant role. One fact about Γ(s) is that it runs off to infinity when s is any integer ≤ 0. (Such a singularity is said to be a “pole” of the function.) Suppose first that s = 1. ζ(1) and Γ(0) both have poles, but because one is in the numerator and the other is in the denominator on the right hand side of the functional equation, they cancel each other out. Γ(1/2) = √π, so as a result ζ(0) is finite and nonzero.

Now suppose s is an odd positive integer ≥ 3. Then Γ(s/2) and ζ(s) are both finite, but there is a pole corresponding to Γ(1-s) in the denominator, so the functional equation shows ζ(-2k) is zero when s = 2k+1 and k is an integer ≥ 1. These values of s = -2k are the only real zeros of ζ(s), as well as the only zeros when Re(s) ≤ 0. They are called the “trivial zeros” of the zeta function. (The Euler product representation of ζ(s) for Re(s) > 1 ensures there are no zeros in that part of the complex plane.)

Finally suppose s = 2k is an even positive integer. Then both Γ(s) and Γ((1-s)/2) are finite and nonzero, as is ζ(s). So ζ(1-2k) is finite and nonzero for integers k ≥ 1. Interestingly enough, those values are rational numbers, which we can write like so:

ζ(1-2k) = -B

_{2k}/ 2k

for k ≥ 1. We write the values in that form, because B_{n} are rather special rational numbers, called Bernoulli numbers, after Jacques Bernoulli (1654-1709) who introduced them. Bernoulli numbers are defined as coeffiecients in a certain power series (the “generating function”):

xe

^{x}/ (e^{x}– 1) = ∑_{0≤n<∞}B_{n}x^{n}/ n!

(The expression x / (e^{x} – 1) can also be used as a generating function to define Bernoulli numbers, and the only difference is in the sign of B_{1}. This fact isn’t immediately obvious, and it implies very interesting recurrence relationships among the Bernoulli numbers.)

The functional equation allows us to give an expression for ζ(2k) as well, although Euler had already discovered the formula. It is:

ζ(2k) = (-1)

^{k+1}B_{2k}(2π)^{2k}/ (2(2k)!)

Unfortunately, the functional equation makes the values ζ(2k+1) totally obscure. It is only relatively recently (1978) that ζ(3) was proven to be irrational (by R. Apéry) – in spite of Euler’s dilligent efforts to compute the ζ(2k+1) numerically for many values of k. We make a big deal about these expressions involving the Bernoulli numbers, since these numbers have connections with diverse other areas of mathematics, including a big role in Fermat’s Last Theorem. We’ll say a bit more about them later.

However, we haven’t yet come to one more important result Riemann stated in his 1859 paper, though he gave no proof of it. (Riemann was the sort of mathematician who “knew” more than he could rigorously prove, as his famous hypothesis shows.) This result is the most important as far as the distribution of prime numbers is concerned. It is an entirely different product formula for ζ(s), which is:

ζ(s) = -e

^{bs}[s(1-s)Γ(s/2)]^{-1}∏_{ρ∈Z}(1-s/ρ)e^{s/ρ}

In this formula b = log(2π) – 1 – γ/2, where γ is Euler’s constant, and Z is the set of zeros of ζ(s) in the so-called “critical strip”, {s∈**C**: 0≤Re(s)≤1}, the portion of the complex plane lying between the lines where Re(s)=0 and Re(s)=1. The complex numbers in Z are the “non-trivial” zeros of ζ(s). It follows from the functional equation of ζ(s) that the zeros are symmetrically distributed around the line Re(s) = 1/2. That is, if ρ = 1/2 + σ + iτ ∈ Z, then so is 1/2 – σ + iτ. The Riemann hypothesis is the conjecture that all members of Z are of the form ρ = 1/2 + iτ.

This formula is rather like the expression for a polynomial f(s), which can be written (up to a constant factor) as

f(s) = ∏

_{ρ}(1-s/ρ)

where the (finite) product is over the zeros of f(s) (assuming no ρ = 0 – if s=0 is also a zero of f(s), just throw in a factor of s^{k} for some integer k > 0, where k is the multiplicity of 0 as a root.)

Given that there are in fact two product formulas for ζ(s), that one involves the prime numbers, and that the other involves the zeros of ζ(s), it’s hard to avoid the suspicion that there is some relationship between the distribution of the primes and the distribution of the zeros. We’ll be more specific about this shortly.

It turns out that many complex analytic functions have a product formula of the above sort, provided they are sufficiently well-behaved. Weierstrass first proved the existence of this kind of representation, and in 1893, just three years before he proved the prime number theorem, Hadamard (motivated by his work on the prime number theorem) proved a more general result, which as a corollary proved the product formula that Riemann conjectured for ζ(s). Such products are now called Hadamard products.

From this representation, together with the functional equation of ζ(s) and some properties of its Dirichlet series, Riemann claimed an asymptotic formula for the number of zeros whose imaginary parts were less than some given value T. More precisely, he claimed

#({ρ∈Z: 0 < Im(ρ) < T}) ∼ (T/2π)log(T/2π) – T/2π

(The notation #(…) means the number of elements in the set.) Zeros in Z are symmetric about the real axis (the line Im(s) = 0), so it isn’t necessary to count zeros with negative imaginary part as well.

Riemann didn’t give a proof of this formula, and it was not rigorously proven until 1905 (by H. von Mangoldt). Nevertheless, he went further and also claimed

#({ρ∈Z: 0 < Im(ρ) < T and Re(ρ) = 1/2}) ∼ (T/2π)log(T/2π) – T/2π

as well. This isn’t quite as strong as saying all zeros are on the line Re(s) = 1/2, just that “most” of them are. Nevertheless, even this weaker result has never yet been proven. As for the strongest possible result, that all zeros lie on the line Re(s) = 1/2, Riemann simply says it is “very likely” true.