John Baez
Zeta functions

This is a draft of a paper by John Baez and James Dolan. For a popularized treatment, start with week300 of This Week’s Finds in Mathematical Physics.

Contents

Introduction

In this paper we begin categorifying the theory of zeta functions. As a precursor, we must understand how to get a Dirichlet series from a species. This is a very nice counterpart to the usual recipe for computing a formal power series from a species, namely its generating function (Joyal).

Briefly, a species is any type of structure that one can put on finite sets: for example, a coloring, or ordering, or tree structure. Suppose FF is some such structure. If we denote the set of FF-structures on the nn-element set by F(n)F(n), the generating function of FF is the formal power series

n0F(n)n!x n. \sum_{n \ge 0} \frac{|F(n)|}{n!} x^n \, .

On the other hand, the Dirichlet series associated to FF is

n1F(n)n!n s. \sum_{n \ge 1} \frac{|F(n)|}{n!} n^{-s} \, .

Far from being ad hoc tricks, the generating function and the Dirichlet series fit into a nice unified theory, which we shall explain here.

The most famous of all Dirichlet series is the Riemann zeta function:

ζ(s)= n1n s. \zeta(s) = \sum_{n \ge 1} n^{-s} \, .

This comes from a species we call the Riemann species, ZZ. A ZZ-structure on a finite set is a way of making it into a semisimple commutative ring — that is, a product of finite fields.

The Riemann zeta function has many generalizations, notably the Hasse-Weil zeta function. This sort of zeta function is usually defined for any projective variety defined over the integers. It is easier to work in still more generality, starting from any functor

S:CommRingSet S : \Comm\Ring \to \Set

that preserves products and takes finite fields to finite sets. Any such functor gives a species which we call the Hasse–Weil species, Z SZ_S. A Z SZ_S-structure on a finite set is defined to be a way of making it into a finite semisimple commutative ring, say kk, and then picking an element of S(k)S(k). The Dirichlet series of the Hasse-Weil species is

n1Z S(n)n!n s. \sum_{n \ge 1} \frac{|Z_S(n)|}{n!} n^{-s} \, .

We shall prove that this equals the usual Hasse-Weil zeta function, which is defined as a product over primes:

pexp( n>0S(𝔽 p n)np ns) \prod_p exp \left( \sum_{n \gt 0} \frac{|S(\mathbb{F}_{p^n})|}{n} p^{-n s} \right)

where 𝔽 p n\mathbb{F}_{p^n} is the field with p np^n elements.

To illustrate how this works, consider a classic example of a Hasse-Weil zeta function: the Dedekind zeta function of an algebraic number field. Given such a field, let RR be its ring of algebraic integers.

Starting from any such ring we obtain a functor from commutative rings to sets, which sends any commutative ring kk to the set of homomorphisms from RR to kk. As described above, this functor gives a Hasse–Weil species. Let us call this species Z RZ_R. Unravelling the constructions we have described, it is easy to see that a Z RZ_R-structure on a finite set is a way of making that set into a semisimple commutative ring and choosing a homomorphism from RR to that ring. By definition, the Dirichlet series of this species is

n1Z R(n)n!n s. \sum_{n \ge 1} \frac{|Z_R(n)|}{n!} n^{-s} \,.

But in fact, this equals the usual Dedekind zeta function, namely

IR/I s \sum_{I} {|R/I|}^{-s}

where II ranges over all ideals of RR, and R/I|R/I| is the cardinality of the quotient ring.

Two Examples

Before plunging into the general theory, let us consider a couple of examples of Hasse–Weil species and their zeta functions. We will tackle these in a brute-force way and make only slight progress. Everything will become easier later, after we have introduced more technology. Still, this first attempt is amusing and perhaps instructive.

First take R=R = \mathbb{Z}. In this case, a Z RZ_R-structure on a finite set is a way of making that set into a commutative semisimple ring and choosing a homomorphism from \mathbb{Z} to that ring. But there is always exactly one such homomorphism, so Z RZ_R-structure is just a way of making a finite set into a commutative semisimple ring. In other words, Z RZ_R is the Riemann species. We call this species ZZ for short.

Let us count the ZZ-structures on an nn-element set for a few small values of nn. To do this, we can start by classifying finite semisimple commutative rings, with the help of two facts:

  • By the Artin–Wedderburn theorem, a finite semisimple commutative ring is the same as a finite product of finite fields.

  • There is one field with qq elements, denoted 𝔽 q\mathbb{F}_q, when qq is a power of a prime number, and none otherwise.

This lets us classify the semisimple commutative rings with nn elements. For example:

  • There is none when n=0n = 0.

  • There is one when n=1n = 1: the ring with one element. (This is the empty product of finite fields.)

  • There is one when n=2n = 2: 𝔽 2\mathbb{F}_2.

  • There is one when n=3n = 3: 𝔽 3\mathbb{F}_3.

  • There are two when n=4n = 4: 𝔽 4\mathbb{F}_4 and 𝔽 2×𝔽 2\mathbb{F}_2 \times \mathbb{F}_2.

  • There is one when n=5n = 5: 𝔽 5\mathbb{F}_5.

  • There is one when n=6n = 6: 𝔽 2×𝔽 3\mathbb{F}_2 \times \mathbb{F}_3.

  • There is one when n=7n = 7: 𝔽 7\mathbb{F}_7.

  • There are three when n=8n = 8: 𝔽 8\mathbb{F}_8, 𝔽 2×𝔽 4\mathbb{F}_2 \times \mathbb{F}_4 and 𝔽 2×𝔽 2×𝔽 2\mathbb{F}_2 \times \mathbb{F}_2 \times \mathbb{F}_2.

Next, how many ways are there to make an nn-element set into a ring isomorphic to one on our list? For starters: how many ways are there to make an nn-element set into a ring isomorphic to some fixed nn-element ring, say kk? Each bijection between the set and the ring kk gives a way to do this, but not all of them give different ways: two bijections differing by an automorphism of kk give the same way. So, the answer is n!/Aut(k)n!/|Aut(k)|.

To go further, we need to know that if q=p mq = p^m for some prime pp, then

Aut(𝔽 q)/m,Aut(\mathbb{F}_q) \cong \mathbb{Z}/m \, ,

a cyclic group generated by the Frobenius automorphism

F: 𝔽 q 𝔽 q x x p. \array{ F : &\mathbb{F}_q &\to& \mathbb{F}_q \\ & x &\mapsto & x^p \, . }

More generally, if we have a finite product of finite fields, its automorphisms all come from automorphisms of the factors together with permutations of like factors. So, for example, 𝔽 2×𝔽 4\mathbb{F}_2 \times \mathbb{F}_4 has 2 automorphisms (coming from automorphisms of the second factor), while 𝔽 2×𝔽 2×𝔽 2\mathbb{F}_2 \times \mathbb{F}_2 \times \mathbb{F}_2 has 6 (coming from permutations of the factors).

Now we can count how many ways there are to make an nn-element set into a commutative semisimple ring:

  • For n=0n = 0 there are 00 ways.

  • For n=1n = 1 there is 1!/1=1!1!/1 = 1! ways.

  • For n=2n = 2 there are 2!/1=2!2!/1 = 2! ways.

  • For n=3n = 3 there are 3!/1=3!3!/1 = 3! ways.

  • For n=4n = 4 there are 4!/2+4!/2=4!4!/2 + 4!/2 = 4! ways.

  • For n=5n = 5 there are 5!/1=5!5!/1 = 5! ways.

  • For n=6n = 6 there are 6!/1=6!6!/1 = 6! ways.

  • For n=7n = 7 there are 7!/1=7!7!/1 = 7! ways.

  • For n=8n = 8 there are 8!/3+8!/2+8!/6=8!8!/3 + 8!/2 + 8!/6 = 8! ways.

From this evidence, one might boldly guess there are always n!n! ways. In fact, though it is hard to see why from our work so far, this guess is correct for all nn. So, the Dirichlet function of the Riemann species is the Riemann zeta function:

n1n!n!n s=ζ(s), \sum_{n \ge 1} \frac{n!}{n!} n^{-s} = \zeta(s) \, ,

Next take R=[i]R = \mathbb{Z}[i] to be the Gaussian integers. Then a Z RZ_R-structure on a finite set is a way of making that set into a commutative semisimple ring and choosing a square root of 1-1 in this ring. Let us count the Z RZ_R-structures on an nn-element set for a few small values of nn.

For starters, choosing a square root of 1-1 in a product of finite fields is the same as choosing a square root of 1-1 in each factor. So, how many square roots of 1-1 does 𝔽 q\mathbb{F}_q have, where q=p mq = p^m is a prime power? The answer depends on the prime pp, and number theorists have names for 3 cases:

  • Split: if p1mod4p \equiv 1 \, mod \, 4, then there are 2 square roots of 1-1 in 𝔽 p m\mathbb{F}_{p^m} for all m1m \ge 1.

  • Inert: if p3mod4p \equiv 3 \, mod \, 4, then there is no square root of 1-1 in 𝔽 p m\mathbb{F}_{p^m} when mm is odd and 2 when mm is even.

  • Ramified: if p=2p = 2, then there is 1 square root of 1-1 in 𝔽 p m\mathbb{F}_{p^m}.

Combining this information with the tables above, we can count the ways are to make an nn-element set into a commutative semisimple ring equipped with a square root of 1-1:

  • For n=0n = 0 there are 0=0×1!0 = 0 \times 1! ways.

  • For n=1n = 1 there is 1×1!1 \times 1! ways.

  • For n=2n = 2 there are 1×2!1 \times 2! ways.

  • For n=3n = 3 there are 0×3!0 \times 3! ways.

  • For n=4n = 4 there are 0×4!/2+1×4!/2=1×4!0 \times 4!/2 + 1 \times 4!/2 = 1 \times 4! ways.

  • For n=5n = 5 there are 2×5!2 \times 5! ways.

  • For n=6n = 6 there are 0×6!0 \times 6! ways.

  • For n=7n = 7 there are 0×7!0 \times 7! ways.

  • For n=8n = 8 there are 1×8!/3+1×8!/2+1×8!/6=1×8!1 \times 8!/3 + 1 \times 8!/2 + 1 \times 8!/6 = 1 \times 8! ways.

So, when R=[i]R = \mathbb{Z}[i], we obtain the zeta function

ζ R(s)=1 s+2 s+4 s+25 s+8 s+ \zeta_R(s) = 1^{-s} + 2^{-s} + 4^{-s} + 2 \cdot 5^{-s} + 8^{-s} + \cdots

Let us see how this compares to the usual Dedekind zeta function. Recall that this is defined as

IR/I s \sum_{I} {|R/I|}^{-s}

where II ranges over all ideals of RR, and R/I|R/I| is the cardinality of the quotient ring. However, every ideal in a ring of algebraic integers can be uniquely factored into prime ideals, so a standard calculation shows

IR/I s= P11R/P s \sum_{I} {|R/I|}^{-s} = \prod_{P} \frac{1}{1- {|R/P|}^{-s}}

where PP ranges over prime ideals of RR. The prime ideals of R=[i]R = \mathbb{Z}[i] come in three flavors, corresponding to the three kinds of primes mentioned above:

  • Split: if p1mod4p \equiv 1 \, mod \, 4, then the ideal of [i]\mathbb{Z}[i] generated by pp is a product of two different prime ideals P,PP, P', with R/P=R/P=p|R/P| = |R/P| = p.

  • Inert: if p3mod4p \equiv 3 \, mod \, 4, then the ideal P[i]P \subseteq \mathbb{Z}[i] generated by pp is a prime ideal with R/P=p 2|R/P| = p^2.

  • Ramified: if p=2p = 2, then the ideal of [i]\mathbb{Z}[i] generated by pp is the square of a prime ideal PP with R/P=2|R/P| = 2.

These account for all the prime ideals in [i]\mathbb{Z}[i], so the Dedekind zeta function is

P11R/P s = 1(12 s)( p1mod41(1p s) 2)( p3mod41(1p 2s)) = 1(12 s)1(13 2s)1(15 s) 21(17 2s) = (1+2 s+4 s+8 s+)(1+9 s+)(1+25 s+)(1+49 s+) = 1+2 s+4 s+25 s+8 s+ \array{ \prod_{P} \frac{1}{1- {|R/P|}^{-s}} &=& \frac{1}{(1 - 2^{-s})} \; \left(\prod_{p \equiv 1 \, mod \, 4} \frac{1}{(1 - p^{-s})^2} \right) \; \left(\prod_{p \equiv 3 \, mod \, 4} \frac{1}{(1 - p^{-2s})} \right) \\ &=& \frac{1}{(1 - 2^{-s})} \frac{1}{(1 - 3^{-2s})} \frac{1}{(1 - 5^{-s})^2} \frac{1}{(1 - 7^{-2s})} \cdots \\ &=& (1 + 2^{-s} + 4^{-s} + 8^{-s} + \cdots)(1 + 9^{-s} + \cdots)(1 + 2 \cdot 5^{-s} + \cdots)(1 + 49^{-s} + \cdots) \cdot \cdots \\ &=& 1 + 2^{-s} + 4^{-s} + 2 \cdot 5^{-s} + 8^{-s} + \cdots }

So, the zeta function arising from the Hasse–Weil species seems to match the usual Dedekind zeta function. And indeed this is true — though again, the reason is not obvious from our work so far. We need a little theory to see what is really going on.

Dirichlet series

Dirichlet series from tame species

We call the groupoid of finite sets and bijections core(FinSet)core(\Fin\Set), since it is the core of the category FinSet\Fin\Set, whose objects are finite sets and whose morphisms are functions. The category of species or structure types is the functor category

[core(FinSet),Set] [core(\Fin\Set), \Set ]

An object in here, say F:core(FinSet)SetF: core(\Fin\Set) \to \Set, describes a type of structure that you can put on a finite set. For any finite set nn, F(n)F(n) is the collection of structures of that type that you can put on the set nn.

We will mainly be interested in species satisfying a certain finiteness property. So, we define the category of tame species to be the functor category

[core(FinSet),FinSet] [core(\FinSet), \FinSet]

Like the category of species itself, the category of tame species has quite a few interesting monoidal structures. Two of these come from addition and multiplication in the target category FinSet\Fin\Set:

  • the pointwise coproduct, given by

    (F+G)(n)=F(n)+G(n) (F + G)(n) = F(n) + G(n)

    This is usually called addition of species.

  • the pointwise product, given by

    (F×G)(n)=F(n)×G(n) (F \times G)(n) = F(n) \times G(n)

Two more arise via Day convolution from ++ and ×\times in core(FinSet)core(\Fin\Set). Note that while this groupoid does not have coproducts or products, it inherits operations which we may call ++ and ×\times from FinSet\Fin\Set, which does. These in turn give the category of tame species the following two monoidal structures:

  • The operation

    (F CG)(n)= k+k=nF(k)×G(k) (F \cdot_{C} G)(n) = \sum_{k + k' = n} F(k) \times G(k')

    which Aguiar and Mahajan (AguiarMahajan) call the Cauchy product. Be careful of the notation: here nn is a finite set, and we are summing over all ways of writing this set as a disjoint union of two subsets kk and kk'. This is just a lowbrow (and very convenient) way of saying that we sum over equivalence classes of coproduct cocones

    k k n \array{ && k &&&& k' \\ & && \searrow & & \swarrow && \\ &&&& n &&&& }

    where we consider two such cocones, say

    k 1 k 1 n \array{ && k_1 &&&& k'_1 \\ & && \searrow & & \swarrow && \\ &&&& n &&&& }

    and

    k 2 k 2 n \array{ && k_2 &&&& k'_2 \\ & && \searrow & & \swarrow && \\ &&&& n &&&& }

    equivalent if there are isomorphisms k 1k 2k_1 \to k_2, k 1k 2k'_1 \to k'_2 making the diagram built from these isomorphisms and the above two diagrams commute. For example, if nn is 3535-element set, kk is a 1010-element set and kk' is a 2525-element set, there are

    35!10!25! \frac{35!}{10! \, 25!}

    equivalence classes of coproduct cocones. This is a highbrow way of saying that this binomial coefficient counts the ways of writing a 35-element set as a disjoint union of a 10-element set and a 25-element set.

  • The operation

    (F DG)(n)= k×k=nF(k)×G(k) (F \cdot_{D} G)(n) = \sum_{k \times k' = n} F(k) \times G(k')

    which we hereby dub the Dirichlet product. Here we must be even more careful to understand the notation, which is a bit misleading. As before, n,kn, k and kk' denote finite sets, not just numbers. And, dually to above case, we are really summing over equivalence cases of product cones

    n k k \array{ &&&& n \\& && \swarrow && \searrow \\ && k &&&& k' }

    with the equivalence relation defined in a precisely dual way to that above. More concretely, we are summing over ways of organizing the elements of nn into a k×kk \times k' rectangle, but where we count two ways of doing this as equivalent if they differ by permuting the rows and/or permuting the columns. So, for example, if nn is 3535-element set, kk is a 55-element set and kk' is a 77-element set, there are

    35!5!7! \frac{35!}{5! \, 7!}

    equivalence classes of product cones. We see here a funny mutant version of a binomial coefficient.

We are mainly interested here in addition and the Dirichlet product, though they fit into a bigger story. Why these two operations? First of all, they make the category of tame species into a rig category. Second, any tame species

F:core(FinSet)FinSet F: core(\FinSet) \to \FinSet

has a Dirichlet series

F¯(s)= n1F(n)n!n s \overline{F}(s) = \sum_{n \ge 1} \frac{|F(n)|}{n!} n^{-s}

and it is easy to check that

F+G¯=F¯+G¯ \overline{F+G} = \overline{F} + \overline{G}

and more interestingly,

F DG¯=F¯G¯ \overline{F \cdot_{D} G} = \overline{F} \, \overline{G}

Checking the second equation here uses some simple facts about the ‘mutant binomial coefficients’ mentioned above. We believe the appearance of Dirichlet series in number theory is largely due to this fact.

All this should be compared to the much more familiar story involving generating functions of species. Namely, any tame species FF has an (exponential) generating function

F(z)= n0F(n)n!z n |F|(z) = \sum_{n \ge 0} \frac{|F(n)|}{n!} z^n

and it is easy to check that

F+G=F+G |F+G| = |F| + |G|

and

F CG=FG |F \cdot_{C} G| = |F| |G|

(Here checking the second equation uses some simple facts about binomial coefficients.)

In short, the Dirichlet series well-adapted to studying the Dirichlet product of species just as generating functions are well-adapted to the Cauchy product. But, they are just two ways of presenting the same information. The Dirichlet series F¯\overline{F} is obtained from the generating function F|F| by the the change of basis

z nn s. z^n \mapsto n^{-s} \, .

In number theory a closely related change of basis is called the Mellin transform:

(e t) nn sΓ(s) (e^{-t})^n \mapsto n^{-s} \Gamma(s)

and this explains the frequent appearance of Mellin transforms in number theory — though there are some wrinkles here that need to be ironed out. But, regardless of these wrinkles, the basic point is that as long as a species FF obeys F(0)=0|F(0)| = 0, we can freely go back and forth between its generating function and its Dirichlet series.

Dirichlet series from tame stuff types

Species, or structure types, are a special case of stuff types, and we can generalize all our remarks so far to stuff types. Here we just sketch this very briefly; for some more details see (BaezDolan, Morton). A stuff type is a weak 2-functor

F:core(FinSet)GpdF: core(\FinSet) \to \Gpd

which describes a type of stuff that you can put on a finite set. By the Grothendieck construction, we can also think of a stuff type as a functor

p:Xcore(FinSet)p: X \to core(\FinSet)

from some fixed groupoid to the groupoid of finite sets. The idea is that pp is a ‘forgetful functor’ sending any ‘set equipped with FF-stuff’ to its underlying set.

There is a concept of groupoid cardinality generalizing the concept of cardinality for finite sets. Namely, for any groupoid XX, we compute its cardinality X|X| as a sum over isomorphism classes of objects. For each isomorphism class, say [x][x], we take a representative object xx and compute the reciprocal of the order of its automorphism group. Then, we sum these up:

X= [x]1Aut(x) |X| = \sum_{[x]} \frac{1}{|Aut(x)|}

Of course the sum could diverge; we say a groupoid is tame if every object has a finite automorphism group and the sum converges. We say a stuff type

F:core(FinSet)GpdF: core(\FinSet) \to \Gpd

is tame if F(n)F(n) is tame for every finite set nn. A tame stuff type FF has a Dirichlet series

F¯(s)= n1F(n)n!n s \overline{F}(s) = \sum_{n \ge 1} \frac{|F(n)|}{n!} n^{-s}

However, this formula simplifies if we treat our stuff type as a functor p:Xcore(FinSet)p: X \to core(\Fin\Set) . Then we have

F¯(s)= n1p 1(n)n s. \overline{F}(s) = \sum_{n \ge 1} |p^{-1}(n)| n^{-s} \, .

Here p 1(n)p^{-1}(n) is the full inverse image of the nn-element set: the groupoid of all objects xXx \in X such that p(x)p(x) is an nn-element set, and all morphisms in XX between such objects. Now the factor of 1/n!1/n! is built into the groupoid cardinality of p 1(n)p^{-1}(n).

The operations of addition and Dirichlet product — and indeed all the operations listed for species — extend to stuff types, and it is easy to check that for tame stuff types

F+G¯=F¯+G¯ \overline{F+G} = \overline{F} + \overline{G}

and

F DG¯=F¯G¯ \overline{F \cdot_{D} G} = \overline{F} \, \overline{G}

To illustrate usefulness of stuff types, let us recompute the first 8 terms of the Riemann zeta function. This time let us treat the Riemann species as a stuff type, namely the forgetful functor

p:core(FinSSCommRing)core(FinSet) p: core(\FinSSCommRing) \to core(\FinSet)

where FinSSCommRing\Fin\SS\Comm\Ring is the category of finite semisimple commutative rings. The Dirichlet series of the Riemann species is then

F¯(s)= n1p 1(n)n s \overline{F}(s) = \sum_{n \ge 1} |p^{-1}(n)| n^{-s}

where p 1(n)p^{-1}(n) is the groupoid of nn-element semisimple commutative rings. As we shall later see, we always get p 1(n)=1|p^{-1}(n)| = 1. For example:

  • There is one isomorphism class of semisimple commutative rings with nn elements when n=1n = 1: the empty product of finite fields, which is the ring with one element. This has just one automorphism, so

    p 1(1)=11=1.|p^{-1}(1)| = \frac{1}{1} = 1 \, .
  • There is one isomorphism class when n=2n = 2: 𝔽 2\mathbb{F}_2. Thus

    p 1(2)=1Aut(𝔽 2)=11=1.|p^{-1}(2)| = \frac{1}{|Aut(\mathbb{F}_2)|} = \frac{1}{1} = 1 \, .
  • There is one when n=3n = 3: 𝔽 2\mathbb{F}_2. Thus

    p 1(3)=1Aut(𝔽 3)=11=1.|p^{-1}(3)| = \frac{1}{|Aut(\mathbb{F}_3)|} = \frac{1}{1} = 1 \, .
  • There are two when n=4n = 4: 𝔽 4\mathbb{F}_4 and 𝔽 2×𝔽 2\mathbb{F}_2 \times \mathbb{F}_2. Thus:

    p 1(4)=1Aut(𝔽 4)+1Aut(𝔽 2×𝔽 2)=12+12=1.|p^{-1}(4)| = \frac{1}{|Aut(\mathbb{F}_4)|} + \frac{1}{|Aut(\mathbb{F}_2 \times \mathbb{F}_2)|} = \frac{1}{2} + \frac{1}{2} = 1 \, .
  • There is one when n=5n = 5: 𝔽 5\mathbb{F}_5. Thus

    p 1(5)=1Aut(𝔽 5)=11=1.|p^{-1}(5)| = \frac{1}{|Aut(\mathbb{F}_5)|} = \frac{1}{1} = 1 \, .
  • There is one when n=6n = 6: 𝔽 2×𝔽 3\mathbb{F}_2 \times \mathbb{F}_3. Thus

    p 1(6)=1Aut(𝔽 2×𝔽 3)=11=1.|p^{-1}(6)| = \frac{1}{|Aut(\mathbb{F}_2 \times \mathbb{F}_3)|} = \frac{1}{1} = 1 \, .
  • There is one when n=7n = 7: 𝔽 7\mathbb{F}_7. Thus

    p 1(7)=1Aut(𝔽 7)=11=1.|p^{-1}(7)| = \frac{1}{|Aut(\mathbb{F}_7)|} = \frac{1}{1} = 1 \, .
  • There are three when n=8n = 8: 𝔽 8\mathbb{F}_8, 𝔽 2×𝔽 4\mathbb{F}_2 \times \mathbb{F}_4 and 𝔽 2×𝔽 2×𝔽 2\mathbb{F}_2 \times \mathbb{F}_2 \times \mathbb{F}_2. Thus

    p 1(8)=1Aut(𝔽 8)+1Aut(𝔽 2×Aut(𝔽 4))+1Aut(𝔽 2×𝔽 2×𝔽 2)=13+12+16=1.|p^{-1}(8)| = \frac{1}{|Aut(\mathbb{F}_8)|} + \frac{1}{|Aut(\mathbb{F}_2 \times Aut(\mathbb{F}_4))|} + \frac{1}{|Aut(\mathbb{F}_2 \times \mathbb{F}_2 \times \mathbb{F}_2)|}= \frac{1}{3} + \frac{1}{2} + \frac{1}{6} = 1 \, .

The calculations here are fundamentally the same as those we did before, but they are cleaner: no cancellation of factorials is needed.

Dirichlet exponentiation

Given a tame species FF with F(0)=0F(0) = 0, there is a tame species exp D(F)exp_D(F) called the Dirichlet exponential such that an exp D(F)exp_D(F)-structure on a finite set is the same as a way of writing that set as an unordered product of finite sets and putting an FF-structure on each factor. This has the property that

exp D(F)¯=exp(F¯) \overline{exp_D(F)} = exp(\overline{F})

where on the right we take the exponential of the Dirichlet series of FF, obtaining another Dirichlet series.

This is reminiscent of the ‘usual’ exponential of a species FF, which we dub the Cauchy exponential exp C(F)exp_C(F). This is a tame species such that an exp C(F)exp_C(F)-structure on a finite set is the same as a way of writing that set as an unordered coproduct of finite sets and putting an FF-structure on each summand. The usual exponential has the property that

exp C(F)=exp(F). |exp_C(F)| = exp(|F|) \, .

Like the usual exponential, the Dirichlet exponential generalizes from species to stuff types.

Multiplicative species and stuff types

In number theory, an arithmetic function is a function

f: + f: \mathbb{N}^+ \to \mathbb{C}

An arithmetic function is said to be multiplicative if

f(mn)=f(m)f(n) f(m n) = f(m) f(n)

whenever mm and nn are relatively prime. Since 11 is relatively prime to everything, this forces f(1)=1f(1) = 1. A multiplicative function is determined by its values on prime powers, so we may think of it as a function of isomorphism classes of finite fields.

Any arithmetic function ff gives a Dirichlet series

f^(s)= n1f(n)n s \hat{f}(s) = \sum_{n \ge 1} f(n) n^{-s}

and ff is multiplicative if and only if this Dirichlet series has an Euler factorization:

n1f(n)n s= p(1+f(p)p s+f(p 2)p 2s+) \sum_{n \ge 1} f(n) n^{-s} = \prod_{p} (1 + f(p) p^{-s} + f(p^2) p^{-2s} + \cdots )

where the product is taken over all primes. The most famous case is when f(n)=1f(n) = 1 for all nn; then we get Euler’s formula for the Riemann zeta function:

n1n s= p11p s \sum_{n \ge 1} n^{-s} = \prod_{p} \frac{1}{1 - p^{-s}}

All this is ripe for categorification.

Definition: A species

F:core(FinSet)SetF : core(FinSet) \to Set

is multiplicative if

F= p DF p F = \prod^D_p F_p

where pp ranges over prime numbers and F pF_p is a species with the property that F p(n)F_p(n) is the empty set unless the cardinality of nn is a power of pp, and the one-element set when nn has one element.
Here D\displaystyle{\prod^D} stands for the Dirichlet product. (Note that while this is an infinite product, there is no real problem, since any natural number is a product of finitely many primes.)

It is easy to check that if a tame species FF is multiplicative, so is the arithmetic function

nF(n) n \mapsto |F(n)|

or for that matter

nF(n)/n! n \mapsto |F(n)|/n!

So, the Dirichlet series of FF has an Euler factorization.

The concept of ‘multiplicative species’ easily generalizes to stuff types, and these too have Dirichlet series that admit Euler factorizations.

The Riemann species

The Riemann species and the Riemann zeta function

Definition: The Riemann species is the species ZZ that assigns to any finite set the collection of ways of making that set into a product of finite fields.

Theorem: The Dirichlet series of the Riemann species is the Riemann zeta function:

Z=ζ. |Z| = \zeta \, .

In other words, the number of ways to make an nn-element set into a product of finite fields is n!n!. To prove this we use some lemmas:

Lemma: The Riemann species is multiplicative:

Z p DZ p Z \cong \prod^D_p Z_p

where Z pZ_p is the species that assigns to any finite set the collection of ways of making that set into a product of finite fields of characteristic pp.

Proof: Use the the definition of ‘multiplicative species’ and the fact that any finite field is a field of characteristic pp for some unique prime pp.

Lemma: The species Z pZ_p can be written as a Dirichlet exponential

Z pexp D(F p) Z_p \cong exp_D(F_p)

where F pF_p is the species that assigns to any finite set the collection of ways of making that set into a finite field of characteristic pp.

Proof: Use the definition of ‘Dirichlet exponential’.

Lemma: The species F pF_p has the following Dirichlet series:

F p(s)=p s+p 2s2+p 3s3+=ln(11p s) F_p(s) = p^{-s} + \frac{p^{-2s}}{2} + \frac{p^{-3s}}{3} + \cdots = ln \left(\frac{1}{1 - p^{-s}}\right)

Proof: A finite field of characteristic pp has cardinality p np^n for some n1n \ge 1. All fields of this cardinality are isomorphic, and the automorphism group of any one of these is a cyclic group with nn elements, generated by the Frobenius automorphism

xx p. x \mapsto x^p \, .

So, to count number of different ways to make an p np^n-element set into a finite field, we can just count the group of all permutations of that set, modulo the subgroup that fixes a given finite field structure, obtaining

(p n)!n \frac{(p^n)!}{n}

To get the corresponding coefficient of the Dirichlet series for the species F pF_p, we just divide by (p n)!(p^n)!. So, the Dirichlet series is

F p(s)= n11np ns. |F_p|(s) = \sum_{n \ge 1} \frac{1}{n} p^{-n s} \, .

This is just another way of writing the expression in the statement of the lemma.

Lemma: The species Z pZ_p has the following Dirichlet series:

Z p(s)=11p s. |Z_p|(s) = \frac{1}{1 - p^{-s}} \, .

Proof: Since Z pZ_p is the Dirichlet exponential of F pF_p we have

Z p(s)=exp(F p(s)), |Z_p|(s) = exp(|F_p|(s))\, ,

so this result follows from the previous Lemma.

These lemmas, together with the Euler product formula for the Riemann zeta function, add up to prove the Theorem:

Z(s) = pZ p(s) = p11p s = ζ(s). \array{ |Z|(s) &=& \prod_p |Z_p|(s) \\ &=& \prod_p \frac{1}{1 - p^{-s}} \\ &=& \zeta(s) \, . }

Hasse–Weil species

These days, algebraic geometry is often formulated in terms of schemes. The most popular definition takes a bit of getting used to, but for the purposes of these notes we can work with a simpler and more general concept, namely a functor

S:CommRingSet. S: \Comm\Ring \to Set \, .

For example:

  • We can start with a polynomial equation with integer coefficients, say

    x 3+y 3=z 3 x^3 + y^3 = z^3

    and let S(k)S(k) be the set of solutions of this equation when the variables take values in the commutative ring kk. Any homomorphism kkk \to k' gives a map S(k)S(k)S(k) \to S(k'), and it is easy to check that SS is a functor. We call S(k)S(k) the set of kk-points of SS.

  • More generally, we can specify a functor S R:CommRingSetS_R: \Comm\Ring \to Set by giving a commutative ring RR and letting S R(k)S_R(k) be the set of homomorphisms f:Rkf: R \to k. The previous example is the special case where we take R=[x,y,z]/x 3+y 3=z 3R = \mathbb{Z}[x,y,z]/\langle x^3 + y^3 = z^3 \rangle.

  • More generally, any scheme determines a functor from CommRing\Comm\Ring to SetSet, often called its ‘functor of points’. The previous example is a special case of this: a so-called affine scheme.

Definition: Given a functor S:CommRingSetS: \Comm\Ring \to Set, we define its Hasse–Weil species Z SZ_S as follows: a Z SZ_S-structure on a finite set is a way to make that set into a finite commutative semisimple ring, say kk and then choose an element of S(k)S(k).

Remember that ‘finite commutative semisimple ring’ is just an erudite term for ‘finite product of finite fields’.

Definition: We say a functor S:CommRingSetS: \Comm\Ring \to Set is tame if its Hasse–Weil species is tame. Equivalently, SS is tame if S(k)S(k) is finite whenever kk is a finite commutative semisimple ring.

Whenever SS is the functor of points of some scheme of finite type defined over the integers, it is tame (Serre). We can think of such a scheme as a projective algebraic variety defined over the integers.

Definition: Given a tame functor S:CommRingSetS: \Comm\Ring \to Set we define its zeta function ζ S\zeta_S to be the Dirichlet series of the Hasse–Weil species Z SZ_S:

ζ S(s)= n1Z S(n)n!n s. \zeta_S(s) = \sum_{n \ge 1} \frac{|Z_S(n)|}{n!} \, n^{-s} \, .

Definition: A functor S:CommRingSetS: \Comm\Ring \to Set is multiplicative if it preserves products.

In fact whenever SS is the functor of points of some scheme defined over the integers, it is multiplicative. This is easiest to see for affine schemes.

Proposition: If S:CommRingSetS: \Comm\Ring \to Set is multiplicative, the Hasse-Weil species Z SZ_S is multiplicative:

Z S p DZ S,p Z_S \cong \prod_p^D Z_{S,p}

where a Z S,pZ_{S,p}-structure on a finite set is a way to make that set into a ring kk that is a product of finite fields of characteristic pp and then choose an element of S(k)S(k).

Proof: The proof is analogous to the case of the Riemann species.

Proposition: If S:CommRingSetS: \Comm\Ring \to Set is multiplicative, each species Z S,pZ_{S,p} can be written as the Dirichlet exponential

Z S,pexp p D(F S,p) Z_{S,p} \cong exp_p^D \, (F_{S,p})

where an F S,pF_{S,p}-structure on a finite set is a way to make that set into a field kk of characteristic pp and then choose an element of S(k)S(k).

Proof: The proof is analogous to the case of the Riemann species.

Here we have factored the Hasse–Weil species into Euler factors Z S,pZ_{S,p} and then written each factor as an exponential. We can also reverse the order, writing the Hasse-Weil species as an exponential of some species and then writing this species as a sum. In other words, we have a commuting square of isomorphisms:

(1)Z S p DZ S,p exp D(F S) exp D( pF S,p) pexp D(F S,p) \array{ Z_S & \rightarrow & \prod_p^D Z_{S,p} \\ \downarrow & & \downarrow \\ exp_D(F_S) &\rightarrow & exp_D (\sum_p F_{S,p}) \cong \prod_p \exp_D(F_{S,p}) }

Proposition: If SCommRingSetS \Comm\Ring \to Set is multiplicative, the species Z SZ_S can be written as the Dirichlet exponential

Z Sexp D(F S) Z_S \cong \exp_D(F_S)

where an F SF_S-structure on a finite set is a way to make that set into a field kk and then choose an element of S(k)S(k).

Proof: We use the assumption that Z SZ_S is multiplicative and the fact that every finite semisimple commutative algebra can be written as a product of finite fields in a unique way.

It is then easy to derive the commutative square (1) with the help of the fact that for any sequence of species F iF_i, we have a natural isomorphism

exp D( iF i) i Dexp D(F i). \exp_D(\sum_i F_i) \cong \prod^D_i \exp_D(F_i) \, .

Let us illustrate these concepts with two easy examples:

Example: Affine dd-space, 𝔸 d\mathbb{A}^d. This is the affine scheme corresponding to the commutative ring R=[x 1,,x d]R = \mathbb{Z}[x_1, \dots, x_d]. Here

S(k)=k d S(k) = k^d

for any commutative ring kk. The corresponding Hasse–Weil species, which we denote as Z 𝔸 dZ_{\mathbb{A}^d}, works as follows. A Z 𝔸 dZ_{\mathbb{A}^d}-structure on a finite set is a way to make that set into a semisimple commutative ring, say kk, and then choose a dd-tuple of elements of it. We have already seen from our study of the Riemann species that there are n!n! ways of making an nn-element set into a product of finite fields. So, the number of Z 𝔸 dZ_{\mathbb{A}^d}-structures on an nn-element set is n dn!n^d n!, and the corresponding zeta function is

ζ 𝔸 d(s) = n1n dn s = ζ(sd), \array{ \zeta_{\mathbb{A}^d}(s) &=& \sum_{n \ge 1} n^d n^{-s} \\ &=& \zeta(s-d) \, , }

just a translate of the Riemann zeta function.

Example: A ring of algebraic integers. This is the affine scheme corresponding to a ring RR consisting of the algebraic integers for some algebraic number field. We define a Z RZ_R-structure on a finite set to be a way to make that set into a semisimple commutative ring, say kk, and then choose a homomorphism from RR to kk. We write the corresponding zeta function as ζ R\zeta_R:

ζ R(s)= n1Z R(n)n!n s. \zeta_R(s) = \sum_{n \ge 1} \frac{|Z_R(n)|}{n!} \, n^{-s} \, .

We claim that ζ R(s)\zeta_R(s) equals the usual Dedekind zeta function of our algebraic number field, namely

IR/I s \sum_{I} {|R/I|}^{-s}

where II ranges over all ideals of RR, and R/I|R/I| is the cardinality of the quotient ring.

On the one hand, it is known that the Dedekind zeta function has an Euler factorization into local zeta functions:

(2) IR/I s= pexp( n>0hom(R,𝔽 p n)np ns) \sum_{I} {|R/I|}^{-s} = \prod_p exp \left( \sum_{n \gt 0} \frac{|hom(R,\mathbb{F}_{p^n})|}{n} p^{-n s} \right)

where hom(R,𝔽 p n)|hom(R,\mathbb{F}_{p^n})| is the number of homomorphisms from RR to 𝔽 p n\mathbb{F}_{p^n}.

On the other hand, we have seen that

Z R p Dexp D(F p,R) Z_R \cong \prod^D_p exp_D(F_{p,R})

where an F p,RF_{p,R}-structure on a finite set is a way of making it into a finite field. It follows that

(3)ζ R(s)= pexp( n1F R,p(p n)p n!p ns). \zeta_R(s) = \prod_p exp \left(\sum_{n \ge 1} \frac{|F_{R,p}(p^n)|}{p^n!} p^{-n s} \right) \, .

Comparing (2) and (3), we see that to prove our claim it suffices to show that

p n!nhom(R,𝔽 p n)=F R,p(p n). \frac{p^n!}{n} |hom(R,\mathbb{F}_{p^n})| = |F_{R,p}(p^n)| \, .

By definition F R,p(p n)|F_{R,p}(p^n)| is the number of ways to make an p np^n-element set into a field and choose a homomorphism from RR to this field. But there are p n!/np^n! / n ways to make a p np^n-element set into a field, since there are p n!p^n! bijections between the set and 𝔽 p n\mathbb{F}_{p^n}, and this field has nn automorphisms. So, there are p n!nhom(R,𝔽 p n)\frac{p^n!}{n} |hom(R,\mathbb{F}_{p^n})| ways to make a p np^n-element set into a field and then choose a homomorphism from RR to this field.

The Hasse–Weil Zeta Function

We are now ready to prove the main theorem. All the real work has been done, since all the essential ideas appear already for the Dedekind zeta function — our second example above. Indeed, the original definition of the Dedekind zeta function is, from a modern point of view, just a clever repackaging of the Hasse–Weil zeta function. Everything becomes simpler if we work with the Hasse-Weile zeta function itself. We can generalize the usual definition of this function as follows:

Definition: Suppose a functor S:CommRingSetS: \Comm\Ring \to Set is tame and multiplicative. Define its Hasse–Weil zeta function to equal

pexp( n>0S(𝔽 p n)np ns). \prod_p exp \left( \sum_{n \gt 0} \frac{|S(\mathbb{F}_{p^n})|}{n} p^{-n s} \right) \, .

Theorem: Suppose a functor S:CommRingSetS: \Comm\Ring \to Set is tame and multiplicative. Then the Dirichlet series of its Hasse–Weil species

ζ S(s)= n1Z S(n)n!n s \zeta_S(s) = \sum_{n \ge 1} \frac{|Z_S(n)|}{n!} \, n^{-s}

is equal to its Hasse–Weil zeta function.

Proof: By (1), in this situation Z SZ_S can be written as a product of Dirichlet exponentials

Z S pexp D(F S,p) Z_S \cong \prod_p \exp_D(F_{S,p})

so taking Dirichlet series we have

ζ S(s)= pexp( n>0F S,p(n)p n!p ns). \zeta_S(s) = \prod_p \exp \left( \sum_{n \gt 0} \frac{|F_{S,p}(n)|}{p^n!} p^{-n s} \right) \, .

It therefore suffices to show that

p n!nS(𝔽 p n)=F R,p(p n). \frac{p^n!}{n} |S(\mathbb{F}_{p^n})| = |F_{R,p}(p^n)| \, .

By definition F S,p(p n)|F_{S,p}(p^n)| is the number of ways to make an p np^n-element set into a field, say kk, and choose an element of S(k)S(k). But as we have seen, there are p n!/np^n! / n ways to make a p np^n-element set into a field. So, there are p n!nS(𝔽 p n)\frac{p^n!}{n} |S(\mathbb{F}_{p^n})| ways to make a p np^n-element set into a field kk and then choose an element of S(k)S(𝔽 p n)S(k) \cong S(\mathbb{F}_{p^n}).

Acknowledgements

We thank Matthew Emerton and other denizens of the n-Category Café for help with Hasse-Weil zeta functions.

References

(AguiarMahajan) Marcelo Aguiar and Swapneel Mahajan, Monoidal Functors, Species and Hopf Algebras, to be published by Cambridge U. Press. Also available online at http://www.math.tamu.edu/~maguiar/a.pdf.

(BaezDolan) John Baez and James Dolan, From finite sets to Feynman diagrams, in Mathematics Unlimited - 2001 and Beyond, vol. 1, edited by Björn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50.

(Joyal) André Joyal, Une théorie combinatoire des séries formelles, Adv. Math 42 (1981), 1–82.

André Joyal, Foncteurs analytiques et espèces des structures, in Combinatoire Énumérative, Lecture Notes in Mathematics 1234, Springer, Berlin, 1986, pp. 126–159.

(Morton) Jeffrey Morton, Categorified algebra and quantum mechanics, Theory and Applications of Categories, 16 (2006), 785-854.

(Serre) Jean-Pierre Serre, Zeta and LL functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper and Row, 1965, pp. 82–92.

Revised on June 12, 2012 10:55:43 by Andrew Stacey? (129.241.15.200)