The prospects for generalizing the Riemann hypothesis don’t seem good. It isn’t at all clear what stronger conclusions might be true – since some which have been considered have turned out to be false. But then, everything discussed here so far has been related pretty closely to the single function ζ(s) and simple variations. It turns out that there’s a large class of functions which are like ζ(s) in many respects – including a strong relevance to number theory. The fact that ζ(s) is the prototype of a whole class of functions that are very important in number theory is another reason for the celebrated status of the Riemann hypothesis.
Most of these more general functions can be expressed as Dirichlet series or something similar. They usually can also be expressed in terms of a product like Euler’s and satisfy a functional equation like that of ζ(s). The numerical values of such functions at special points (and their “residues” at poles) have particular significance in terms of algebraic and arithmetic objects that the functions are associated with. And they are also expected to satisfy a condition similar to Riemann’s hypothesis regarding the location of their zeros.
With number theory in mind, there is a generalization of a quite important kind we might consider. In “ordinary” number theory the numbers that are of principal concern are rational numbers (the set of which is denoted by ℚ) and the integers (ℤ). The theory of rational numbers and integers is quite rich and interesting. A great deal of it is concerned with properties of prime numbers. Many natural questions in number theory can be expressed in terms of integers and rationals.
Not all questions, however. Any time one wants to deal with Diophantine equations that are not “linear”, even simple ones such as x2 – d = 0, it is necessary to go beyond the rationals and integers to consider “algebraic” numbers, which are, roughly speaking, any solution of a polynomial equation having rational numbers as coefficients. This involves the vast mathematical subject of “algebraic number theory”, of which “ordinary” number theory — the theory of rational numbers and integers — is only a small part.
Algebraic number theory is a huge and deep subject, with many difficult open questions of its own. We go into more detail on this subject elsewhere, but there are straightforward analogues of ζ(s) that play a large role in the theory and are relevant to mention here.
Many of the important questions of algebraic number theory revolve around the properties of suitable analogues of prime numbers. It is not hard to define the meaning of “integers” within the class of all algebraic numbers. However, in general algebraic number theory one does not usually have a set of numbers (even algebraic numbers) in terms of which any arbitrary algebraic integer can be expressed uniquely as a product of “prime” numbers of some sort. One must instead work with certain sets of algebraic numbers called “ideals”. This makes the theory much more complicated. It is therefore striking that a fairly simple analogue of the Riemann zeta function does exist. The analogue is the Dedekind zeta function, after Richard Dedekind (1831-1916) who did much to develop algebraic number theory, and was a close friend of Riemann and his first biographer as well.
To talk about Dedekind zeta functions we need a little terminology. First, a “field” is a collection of mathematical objects (such as numbers or functions) for which there are concepts analogous to the addition and multiplication of ordinary numbers. There are certain axiomatic rules which specify how addition and multiplication should behave. For instance, there is the “distributive” rule of multiplication with respect to addition: a(b + c) = ab + ac for any a, b, and c belonging to the field. The rational numbers ℚ form a field, as do the real numbers ℝ and the complex numbers ℂ.
Fields can contain, and be contained in, other fields. For present purposes we are concerned with fields K that contain the rationals ℚ. Symbolically, ℚ ⊆ K. In this case, one says that ℚ is a subfield of K, and K is an extension of ℚ. An “algebraic number” is simply any root of any polynomial f(x) whose coefficients are rational numbers in ℚ, where polynomials are expressions of the form
f(x) = anxn + an-1xn-1 + … + a1x + a0 with ai ∈ ℚ for 0 ≤ i ≤ n
The numbers ai are the coefficients, and x is referred to as an “indeterminate”. The set of all such polynomials is denoted by ℚ[x]
To say that the algebraic number α is a “root” of f(x) means just that f(α) = 0. If n is the smallest integer such that an ≠ 0, while am = 0 if m > n, n is said to be the “degree” of f(x). An algebraic number α is an “algebraic integer” if there is a polynomial f(x) of degree n where an = 1, and all of the other coefficients are ordinary integers (elements of ℤ). It isn’t hard to show that the subset of an extension K ⊇ ℚ consisting only of algebraic integers is what is called a “ring”, which is a very much like a field, except that its members do not necessarily have multiplicative inverses (i. e. reciprocals) that are also algebraic integers (although the inverses are still algebraic numbers in K). This ring is called the ring of algebraic integers of K/ℚ. (Which is read “K over Q”.) It is this ring which generalizes the ring ℤ of ordinary integers.
In general, an algebraic number α can be a root of infinitely many polynomials, but there is always a polynomial of smallest degree n whose leading coefficient an = 1. This is called the “minimal” polynomial of α. There is a function called the “norm” that maps all algebraic numbers of an extension field K ⊇ ℚ to ℚ. If K is the smallest extension field of ℚ that contains α and all other complex roots of the minimal polynomial f(x) of α (known as the “splitting field” of f(x)), then the norm of α is simply the constant term of f(x), i. e. f(0). It is denoted by NK/ℚ(α) and it is a member of ℚ by definition. If α is an algebraic integer, its norm is an ordinary integer.
Now, as mentioned, the ring of integers R of some K ⊇ ℚ doesn’t necessarily have the property that all its members are products of “primes” in a unique way. This fact makes arithmetic in R much messier than arithmetic in ℤ. Fortunately, algebraists in the 19th century discovered that it is possible to define certain subsets of R called “ideals” that have a property analogous to unique factorization. An ideal I ⊆ R is itself a ring with the further property that all products of elements of I by elements of R are members of I. (Symbolically, IR ⊆ I.) An important type of ideal is a “principal” ideal, which consists of all products of an algebraic integer and other elements of the ring, and is usually written as αR or (α). Although ideals are not numbers, they play an analogous role in the theory. For instance, it is possible to define the norm of an ideal, and the value of the norm is an ordinary integer.
Happily, unique factorization can be recovered for rings of integers as far as ideals are concerned. That is, there are ideals, called prime ideals, such that any ideal in the ring can be expressed as a “product” of prime ideals in a unique way. The way that an algebraic integer α factors in the ring can be expressed in terms of the prime ideals that are factors of the principal ideal (α). Most of algebraic number theory is devoted to describing the prime ideals of a given ring of algebraic integers in greater detail.
One of the key tools used in algebraic number theory is the Dedekind zeta function, which we now have enough terminology to define. The zeta function of the algebraic number field K ⊇ ℚ is given by
ζK(s) = ∑A (NK/ℚA)-s
where the sum runs over all the ideals of the ring of integers of K, and NK/ℚA is the norm of A. Note that this is in fact a Dirichlet series, because the norm is defined so that it is an integer. If K = ℚ, ζK(s) is simply the familiar Riemann zeta function.
Many of the properties of ζ(s) carry over to ζK(s). In particular, precisely because one has unique factorization of ideals, there is a product formula:
ζK(s) = ∏P (1 – 1/(NK/ℚP)s)-1
where the product runs over all prime ideals P.
ζK(s) can also be written as an ordinary Dirichlet series:
ζK(s) = ∑1≤n<∞ f(n)/ns
where f(n) is the number of ideals of K that have norm equal to n. Clearly, f(n) is a key number theoretic function in the theory, and ζK(s) encodes information about it.
The similarities to ζ(s) don’t stop there. There are many others:
- The series and product expressions for ζK(s) converge in the half plane Re(s) > 1, and the function is holomorphic there.
- ζK(s) has a pole at s = 1.
- ζK(s) has a simple functional equation involving the Γ function.
- ζK(s) has zeros at s = -2k and sometimes (depending on the field K) at s = -2k-1 for integers k ≥ 1.
- All other zeros of ζK(s) lie in the critical strip 0 ≤ Re(s) ≤ 1.
- ζK(s) has no zeros with Re(s) = 0, and in fact ζK(s) ≠ 0 if Re(s) ≥ 1 – A/(n log(|T|)), for sufficiently large T = Im(s), a constant A, and n the degree of the extension K over ℚ.
There is even an analogue of the Riemann hypothesis: it is conjectured that all zeros of ζK(s) with Re(s) > 0 lie on the line Re(s) = 1/2. It isn’t known, however, whether in general ζK(s) has zeros with Im(s) = 0 in the critical strip. (ζ(s) doesn’t, of course.) Unsurprisingly, the status of this extended Riemann hypothesis is even more mysterious than that of its special case for ζ(s).
Naturally, the precise characteristics of ζK(s) depend on the specific field K. Indeed, important quantities related to K can be expressed in terms of ζK(s). For instance, the very important “ideal class number” of K, which roughly expresses how badly the ring of integers of K fails to have all of its ideals be principal ideals, enters into the formula for the “residue” of ζK(s) at its pole at s = 1.
The class number is number of elements of the “ideal class group” of the field. This group consists of classes of ideals within the set of all ideals of the ring of integers of K. We won’t attempt to explain that idea more here – it’s part of the very deep theory called “class field theory – except to note that Dedekind zeta functions can also be defined for each of these ideal classes, by letting the sum which defines the function run only over those ideals which are members of a given class:
ζ(s; H) = ∑A∈H (NK/ℚA)-s
for some ideal class H. There are h such classes, where h is the ideal
class number, and
ζK(s) = ∑1≤i≤h ζK(s; Hi)
It is also possible to express ζK(s) in terms of a product of ζ(s) and functions called “L-functions”, the simplest case of which comprises Dirichlet L-functions, which we’ll look at next.