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Chapter 3 Enumeration Problems
CHAPTER 3
ENUMERATION PROBLEMS
In many cases, particularly in the study of arithmetical semigroups associated with categories, one may interpret the (-formulae studied in the previous chapter as the answers to certain types of enumeration questions. In the context of arithmetical categories, those questions often also have an intrinsic character essentially independent of the present number-theoretical formulation. A major problem to be considered in this chapter is that of enumerating the isomorphism classes of objects in a given arithmetical category. It will be seen below that the Euler product formula for the zeta function provides a basic stepping-stone towards the solution of this problem, and that it leads to complete algebraic solutions for quite a number of specific categories of interest.
§ 1. A special algebra homomorphism
The reader with some interest in algebra may have noticed quite early that much of our discussion of the Dirichlet algebra Dir (G) of an arithmetical semigroup G would be largely unaffected if complex-valued functions were replaced by functions f: G -k, where k is a general field with a real valuation I 1 (say). This would especially be the case if k were a field of characteristic 0 and one could factorize the norm mapping on G through k and its valuation I I. In this way, one would obtain a generalized Dirichlet algebra Dir (G, k), whose properties might perhaps be worth studying in detail. In this book, it will not be necessary to go further into the above possibility, but instead it will be useful to consider the possibility of assuming slightly less about the initial semigroup G. For present purposes, it will be sufficient to withdraw the requirement of unique factorization and to consider a semigroup H of the following type: Let H denote a commutative semigroup with identity element 1, with the property that every element aEH has only a finite number of divisors in H, and such that there
CH. 3. §1.
A SPECIAL ALGEBRA HOMOMORPHISM
55
exists a real-valued norm mapping I I on H satisfying the conditions: (i) 111=1, and lal>1 for a,cIEH, (ii) labJ = lallbJ for all a, bE H, (iii) the total number NIl (x ) of elements aEH of norm jal.=2x is finite, for each real x e-O. Under these assumptions, let Dir (H) denote the set of all complexvalued functions on H. One may then imitate the earlier treatment of Dir (G) when G is an arithmetical semigroup, and in the first place make Dir (H) into a complex algebra in the obvious similar fashion. Elements of Dir (H) may be represented as formal Dirichlet series over H, and one may define a norm I II on Dir (H) in the same general way as in the discussion of Dir (G). In that case, this 'Dirichlet algebra' for H will retain many of the properties of Dir (G), although it may perhaps not be an integral domain and I I may only be a pseudo-valuation (i.e., have the properties of I II on Dir (G) given in Proposition 2.1.1, except that Ilf(z)g(z)11 = ~ II filII gil without equality necessarily holding in all cases). In particular, these principles may be applied to the semigroup
IGI = 1m {I I: G - R}, where G is some given arithmetical semigroup. Then there is a continuous identity-preserving algebra homomorphism -: Dir (G) - Dir (lGf)
defined by
fez) =
Z ( Z f(a»)q-Z =
qEIG[
[u!=q
~ f(a)
aEG
la\-Z,
given fE Dir (G). (The norm on IGI is taken to be the identity mapping.) It should be noted that the present Dirichlet series over IGI are still formal series, but it is through these that one may make a convenient transition to ordinary series of analysis when desired at a later stage. For the moment, without making such a transition, it is clear that the earliermentioned problem of enumerating the isomorphism classes in an arithmetical category is intimately related with the associated enumerating function 'G(z) = la[-: = G(q)q-Z
Z
aEG
.z
qEIG[
where G(q) denotes the total number of elements aEG of norm lal=q. Now observe that, since - is a continuous identity-preserving algebra homomorphism, it will preserve pseudo-convergent sums and products. (Note that the concept of pseudo-convergence may be carried over to
56
ENUMERATION PROBLEMS
CH. 3. §l.
Dir (H), and so to Dir (IGI), in the obvious way.) Roughly speaking, this remark may be paraphrased so as to state that the homomorphism "preserves formulae". In particular, the Euler product formula for CO
where IGlo= IGj"'{I}, and P(q) is the number of primes of norm q in G. Given a knowledge of the norms Ipl of the primes pEP, this formula provides a mechanical algebraic procedure for calculating the successive numbers G(q) as qE IGI increases monotonically. In other words, this formula, which (if confusion seems unlikely) we may also refer to as the Euler product formula, in principle provides a complete algebraic solution to the enumeration problem discussed above. Before illustrating this statement by means of examples, we now introduce some further terminology and notation, which seems especially appropriate and convenient for certain types of arithmetical semigroups: If one re-examines the examples listed in Chapter I, it will be noticed that in some cases the norm functions are not entirely "natural", in that they arise by exponentiating certain more immediate functions with slightly different properties (e.g. 'dimension' or 'degree' functions). This suggests the formal concept of an additive arithmetical semigroup, which we define to be a semigroup G with the same unique factorization properties as in Chapter I, § I, but with the norm mapping I I now replaced by a realvalued degree mapping 0 on G such that (i) 0(1)=0, o(p»O for pEP, (ii) o(ab)=o(a)+o(b) for all abc G, (iii) the total number N G '" (x) of elements aE G of degree o (a) ~x is finite, for each real x e-O. Obviously, this concept is only formally different from that of an arithmetical semigroup, since one may clearly interchange norm and degree mappings by the rules
o(a) = loge lal
(aE G),
where c> I is fixed. However, as indicated above, it is nevertheless sometimes more natural or convenient to use the new terminology. It is clear, and we shall take as understood, how one may similarly define an additive arithmetical category (i and its associated additive arithmetical semigroup Grt •
CH. 3. §l.
A SPECIAL ALGEBRA HOMOMORPHISM
57
Now let G denote an additive arithmetical semigroup with degree mapping o. Bya remark at the beginning of Chapter I, § I, it is clear that the above conditions (i) to (iii) are equivalent to (i) and (ii) together with (iii)' the total number 1tG * (x) of primes pE P of degree o(p) -;§x is finite, for each x>O. Now consider the enumerating function (dz) for G that arises by associating with G the norm function
\aJ
=
ca(o)
(aEG),
where c> I is fixed by some suitable choice. If confusion seems unlikely, we may also refer to (G(z) as the zeta/unction of G (or of the arithmetical category (£ to which G is associated, if appropriate) and sometimes later even omit the tilde -. In the present situation, we then have (G(z)
=
Z oio«: = Z
qE IGI
G*(u)c- UZ,
uEa(G) .
where c_uz=(CU)_Z and G*(u) denotes the total number of elements aEG such that (a) = u. Further, the Euler product formula (in the new sense) may be written as
o
(G(z)
= II (I-c-a(p)=)-l = II pEP
(I
_c-UZ)-P*(u),
uEa(G)o
where P*(u) denotes the total number of primes pEP such that oCp)=u, and o(G)o=o(G)""{O}. Now write y" = c- UZ = (c")-Z for uE o(G), so that in Dir (iGI) we have y"yV = (CU)-Z(CV)-Z = (c"+V)-Z = y"+v.
Thus yU behaves symbolically like a "u th power", and we may write
say. The Euler product formula then becomes
A frequent simplification. Before the reader becomes too irritated or discouraged by symbolic "u th powers" and formal sums and products defined over obscure subsets IGI or oCG) of the field of real numbers, we hasten to emphasize that in a large proportion of naturally interesting special
58
CH. 3. §1.
ENUMERATION PROBLEMS
cases (such as those stemming from or included in Chapter I) the natural norm or degree functions take only rational integer values. (If the degree function takes only rational integer values, one can always choose e so that the corresponding norm function takes only rational integer values.) In cases of this kind, there is no need to use generalized Dirichlet algebras of the form Dir (H). For, in the first place, if G has an integer-valued norm mapping I I, then one may work directly with the homomorphism -: Dir (G)
given by
fez) =
Dir (G z )
-+
Z ( Z !(a»)n-
n=l
z
lal=n
[jE Dir(G»).
The enumerating function (zeta function) may now be written as
Z G(n)n-
'G(Z) =
Z
n=l
(with the tilde - omitted), and the Euler product formula has the form 'G(Z) =
II (I
_m-Z)-P(m).
m=l
Secondly, if G has an integer-valued degree mapping corresponding integer-valued norm of the form
lal
=
ko(a)
a,
one may use a
for aEG,
where k » I is an integer. If one substitutes y=k- z in Dir (Gz ), one then obtains an 'enumerating power series' or generating function ZG(Y) =
Z G~ (n)yn,
n=O
and the Euler product formula becomes
II (I _ym)-p*(m). oo
ZG(Y) =
m=l
Over here, one may view Y as an indeterminate in the ordinary algebraic sense; for, Y is algebraically independent over the subalgebra C of Dir (G z ) generated by 1, and so the set of all power series enY' (with complex c.) forms a subalgebra of Dir (Gz) which is isomorphic to the formal power series algebra C [[t J] discussed earlier.
Z:=o
CH. 3. §2. ENUMERATION AND ZETA FUNcnONS IN SPECIAL CASES
59
Bearing the convenience of these simplifications in mind, one may wonder how necessary it is to consider norm or degree mappings that are not integer-valued. One reason for considering the more general case is that one may then derive and use a certain "Normalization Principle" in the context of Part II of this book. Another reason lies in the fact that certain analytical techniques for obtaining asymptotic enumeration formulae (discussed in Part III below) seem to be more conveniently phrased in terms of general real-valued norms or degrees. (One specific situation, in which it becomes useful if not essential to consider degree mappings that are not integer-valued, is considered in the author's paper [12]; this covers certain natural arithmetical categories with integer-valued norm mappings, which nevertheless seem most easily dealt with in terms of associated degree mappings that need not necessarily take only integral values.)
§ 2. Enumeration and zeta functions in special cases At this stage it may be interesting to re-examine the main examples of Chapter I, and see what information the above general discussion yields when applied to such special but important cases. 2.1. Example: The Riemann zeta function. For the semigroup Gz of positive integers, IGzl = Gz and - is the identity automorphism of Dir (Gz). The zeta function of Gz is then
and is called the Riemann zeta function. (In using this terminology, and in referring to the other special zeta functions below, one often thinks of (z) and the other functions as functions of a complex variable z (or y, if one is considering ZG(y»), and questions of ordinary convergence then become relevant. In this chapter, however, we shall continue to work only with formal series.) For the Riemann zeta function, we have the classical Euler product formula
(z)
=
n {(I_p-Z)-l: rational primes pl.
2.2. Example: Euclidean domains. For the arithmetical semigroups of associate classes of non-zero elements of the domains Z [V -I] and Z [V2] discussed in Chapter I, the enumerating functions are special cases of the Dedekind zeta function treated below. However, the information on prime
ENUMERATION PROBLEMS
60
CH. 3. §2.
elements in these domains listed in Chapter I, § I, allows us to write down the Euler products explicitly in terms of products over rational primes. Thus, for the Gaussian integers Z [y' -I] we have
II {(I_p-2z)-l: rational primes p=3(mod4)} xII {(l_p-Z)-2: rational primes p= 1 (mod4)}.
CZ[Y- 11(z) = (1-2- Z ) - 1
(Here, and throughout this section, the tilde over CG(z) is omitted.) Similarly, for the domain Z [Y2] one obtains
CZ l Y21(Z) = (1_2- ) - 1 II {(I_p-2Z)-1: rational primes p= ±3 (mod 8)} Z
xII {(I_p-Z)-2:
rational primes p=
± 1 (mod 8)}.
A very simple but nevertheless quite interesting example is provided by the semigroup of associate classes of non-zero elements of a Galois polynomial ring GF[q, t], i.e. the polynomial ring F[t] in an indeterminate t over the finite Galois field F=GF(q) with q elements. This is an additive arithmetical semigroup G=G[q, t] with degree mapping 8(J)=degf It is easy to see that there are exactly qn elements of degree n in G[q, t]. Hence G[q, t] has the generating function ec
ZGlq,tj(Y) =
Z qnyn =
n=o
(I_qy)-l.
Consider the corresponding Euler product formula
II (1- y"')-P*
ZGlq,tj(Y) =
m=1
We shall now show how this formula leads to an explicit equation for P* (m), and hence to the following proposition. 2.3. Proposition. The total number ofmonic irreducible polynomials ofdegree m in GF[q, t] is
r»
(m) = m- 1
Z p.(d)qm1d,
dim
where p. is the Mobius function on the positive integers. Proof. By the Mobius inversion principle, the proposition will follow once we have established the equation qm =
Z dP*(d).
dim
CH. 3. §2. ENUMERATION AND ZETA FUNCTIONS IN SPEGAL CASES
61
One way to do this is to take 'logarithms' on both sides of the Euler product formula and then compare coefficients of the resulting series. In order to do this, we could first use the usual 'logarithmic series' formula to define log (I +h) whenever h is a power series with o (h) >0, where the order o(h) is defined as in Chapter 2, § 4. After verifying that log is continuous relative to the (y)-adic topology for series of the form I +h, and that log {(l
+ hI) (I + h2 ) }
= log (I
+ hI) + log (l + h 2) ,
one could then proceed with the details indicated. However, familiarity with the algebra of differentiation may perhaps make it slightly simpler to use 'logarithmic derivatives' instead: Firstly, if
define its formal derivative by the usual formula ee
f' =
Z ncny"-l.
n=1
One can verify that the rule f -.f' is continuous relative to the (y)-adic topology and is a derivation, i.e., satisfies the usual rules for differentiation. If f is a unit, we define its logarithmic derivative to be DL(f)=f'/f Then D L is a continuous operation relative to the (y)-adic topology (since formal differentiation, inversion and multiplication are continuous), and Ddfg)
=
Ddf)+DL(g)
if f, g are both units. (If desired, one could define and treat logarithms and logarithmic derivatives of arithmetical functions over a general arithmetical semigroup, in a very similar way.) Now take logarithmic derivatives on both sides of the above Euler product formula. Then oo
DL(ZG[q,tj(y») = q(l_qy)-1 =
Z qn+ly",
n=o
while on the other hand the continuity and the additive property of D L give
Z mP* (m)y'"-I(I_ y'")-1 m=1 ~
DdZG[q,tj(Y») =
ee
= ZmP*(m)y'"-I(I+y'"+y2m+ ... ). m=l
62
ENUMERATION PROBLEMS
CH. 3. §2.
By comparing the coefficients of yn-l one then obtains qn=
Z
m., m-l+rm=n-l
mP*(m)=
Z
m{,+l)=n
mP*(m)=ZdP*(d), din
since the preceding sum allows r = O. This proves the proposition.
0
2.4. Example: The Dedekind zeta function. Let GK denote the arithmetical semigroup of all non-zero integral ideals in a given algebraic number field K. The enumerating function for GK is (,.{z)
=
Z IEG
I/I-z =
K
=
L: K(n)nn=l
Z
,
where K(n) denotes the total number of ideals of norm n in G K • The function (K (z) is known as the Dedekind zeta function of K. The facts about prime ideals in GK mentioned in Chapter I show that the Euler product formula for G K has the form
where pal, ... , pak are the norms of the distinct prime ideals PI' ... , Pk in the decomposition (p)=p1e' ... Pkek. (Since (p) has normp[K:Ql, it follows that e1tl1+ ... +ektlk=[K:Q].) By direct considerations, or by means of this formula and Corollary 2.4.2, one notes that K(n) is a multiplicative function of nEG z. 2.5. Example: Finite abelian groups. For the category .91 of all finite abelian groups, the zeta function may be written as C,,(z) =
Z a(n)n-
n=l
Z
,
where a(n) denotes the total number of isomorphism classes of abelian groups of order n. The discussion of primes in .91 given in Chapter I shows that here the Euler product may be written as a pseudo-convergent double product:
II {(I_p-rz)-l: rational =II II(I_p-rz)-l
C~(z) =
eo
,=1 =
=
p
II (rz),
'=1
primes
p, r ~
I}
CH. 3. §2. ENUMERATION AND ZETA FUNCTIONS IN SPECIAL CASES
63
by the Euler product formula for the Riemann zeta function. Again, these formulae or direct considerations show that a(n) is a multiplicative function in Dir (G z ). Further, the last equation provides an explicit infinite '-formula for a(n) as an element of Dir (G z ). Now consider the subcategory d(p) of all finite abelian p-groups, where p is a fixed rational prime. Since p is now fixed, for some purposes it is natural to regard d(p) as an additive arithmetical category, with degree mapping defined by ()(A) = log, card (A). In that case, d (p) has exactly one prime of degree r for each r = I, 2, .... Therefore the Euler product formula implies that d(p) has the generating function Z..,(p)(Y) =
n (I - y)-I =
,=1
Z
n=O
p(n)y",
where pen) =a(pn) is the total number of isomorphism classes of abelian groups of degree n in the above sense. In fact, for n >0, pen) is a function of the positive integer n familiar to number theorists as the partition function, which may be defined arithmetically as the total number of ways of partitioning n into a sum of positive integers (where the order of the summands is disregarded). For example, p(5) = 7 since
5 = 1+4 = 1+1+3 = 1+1+1+2 = 1+1+1+1+1
= 2+3 = 2+2+ I. It will be seen later that the study of additive arithmetical semigroups is often, as in this case, closely related to 'additive' number theory in the classical sense. (For more information on this and the partition function, see for example Hardy and Wright [I).) 2.5. Example: Semisimple finite rings. Let 6 denote the category of all semisimple finite rings. The enumerating function for 6 (or G(!) may be written as '(!(z) =
Z S(n)n-
Z
,
n=1
where Sen) denotes the total number of isomorphism classes of rings of cardinal n in 6. In this case, the primes in the category are the isomorphism classes of the various simple finite rings, and the discussion in Chapter I,
ENUMERAnON PROBLEMS
64
CH. 3. §2.
§ 2, shows that the corresponding Euler product may be expressed as a pseudo-convergent triple product (e(Z) = =
n {(I_p-rmSZ)-l: rational primes p, r ~ I, m ~ I} Il Il (l_p-rmaz)-l
r~l
p
m~l
=
n (rm
2z),
r~l
m;i!;l
by the Euler product formula for the Riemann zeta function. The function S(n) of the positive integer n is multiplicative, and the last equation provides an explicit infinite (-formula for it as an element of Oir (Gz). Now consider the subcategory 6(p) of all semisimple finite p-rings, where p is now a fixed rational prime. As with .9I(p), it is convenient to regard this category as an additive arithmetical category, with degree mapping defined by o(R) = log, card (R). Here the discussion in Chapter I shows that the primes of 6(p) may be arranged in a double sequence (P rm) (r,m= 1,2,... ), where o(Prm ) = rm 2 • Hence the Euler product has the form Ze(p)(Y) =
n (1-
r~l m~l
Z s(n)y", ~
yml)-l
=
n=O
where s(n) = S(p") is the total number of isomorphism classes of rings of degree n in 6(p), and s(n) may be viewed as a natural generalized partition function of the positive integer n. For small n, it is easy to calculate s(n); for example, s(5)=8. (Tables of values for p(n) and s(n) when n;§50 are provided in Appendix 2.) By the discussion in Chapter I, it is not difficult to verify that the subcategory 6 c of all commutative rings in 6 is 'arithmetically equivalent' to the category .91, in the sense indicated at the end of Chapter I. Hence 6 c and .91 have the same enumerating function. Similarly, the subcategory 6 c(p) of all p-rings in 6 c (for a fixed rational prime p) is arithmetically equivalent to the category .9I(p), and has the same generating function. 2.6. Example: Semisimple fioite-dimensiooal algebras. Consider the category 6(F) of all semisimple finite-dimensional associative algebras over a given
CH. 3. §2. ENUMERATION AND ZETA FUNCTIONS IN SPECIAL CASES
65
field F. Three cases were discussed in Chapter I, § 2, and in each of them 6(F) is an additive arithmetical category with the degree function o(A) = dimA.
Case (i): F is an algebraically closed field. In this case there is exactly one prime of degree m 2 for each m= 1,2, .... Therefore 6(F) has the generating function co
Ze(F)(Y) = II (1- y"'")-1 = m=1
ee
Z
p(2) (n)y",
n=O
where p(2 l (n) is the total number of isomorphism classes of n-dimensional algebras in 6(F), and, for n>O, p(2)(n) may be interpreted arithmetically (in the classical sense) as the total number of partitions of n into a sum of squares. Case (ii): F is a real closed field R. Here, by the discussion in Chapter I, the primes split into three sequences (Pm), (Qm) and (R m), where o(Pm) = m 2 ,
(m = 1,2, ... ).
Therefore ~
Ze(Rl(Y) =
II[(I_y"'")(l_y 2mz)(l_y4m")]-1 m=1
say, where SR (n) enumerates the isomorphism classes of n-dimensional algebras in 6 (R). Case (iii): F is a finite Galois field GF(q). The description of the primes for 6(F) in this case given in Chapter I shows that now the category 6(F) is arithmetically equivalent to the category 6(p) above. Hence 6(F) has the same generating function as 6(p) in this case.
2.7. Example: Compact Lie groups, semisimple Lie algebras, and symmetric Riemannian manifolds. The description given under Examples 2.4 and 2.5 in Chapter I shows that the categories considered there may be regarded as additive arithmetical categories. We leave the computation of the generating functions (correct up to a finite number of factors) as an exercise. 2.8. Example: Pseudo-metrizable finite topological spaces. For illustrative purposes, it is interesting to note that the description given in Chapter I shows that the category ~ of these spaces forms an additive arithmetical category,
66
CH. 3. §2.
ENUMERATION PROBLEMS
if one defines the degree of a space X to be card (X). There is then exactly one prime of degree m for each m = I, 2, .... Hence ~ is arithmetically equivalent to the category d(p) above, and has the same generating function p(n)yn, where p(n) is the partition function.
Z:=o
2.9. Example: Finite modules over a ring of algebraic integers. Finally, consider the category ~ = lYD of all modules of finite cardinal over the ring D of all algebraic integers in a given algebraic number field K. If aD(n) denotes the total number of isomorphism classes of modules of cardinal n in ~, then
=
C;v(Z)
Z
n=l
aD(n)n- Z
and the description of the indecomposable modules in shows that
lY given
in Chapter I
II {(1-(I P n- )- l: prime ideals Pin D, r ~ = II II {(I - !PI-rz)-l: prime ideals P in D}
Cil(z) =
Z
I}
=
r~l
where CK(Z) is the Dedekind zeta function discussed above. It follows that aD(n) is a multiplicative function of nE Gz. In studying aD(n) and the category lY or semigroup G;v in general, it is sometimes convenient to factorize the homomorphism - : Dir (G;v) - Dir (Gz) as follows: First note that there is an identity-preserving and norm-preserving homomorphism 4>: G;v -G K defined by sending the isomorphism class of DIP' (P a prime ideal, r s: I) to P', and extending this rule multiplicatively. Then 4> is actually an epimorphism, and 4> induces a continuous identitypreserving algebra homomorphism 4>*: Dir(G,,) - Dir(G K ) according to the rule f
-+
4> * (j), where
4>*(/) (z) =
Z( Z
IEG K
(M)=1
f(M»)/-z.
(Here M denotes the isomorphism class of the module M in lY.) Since 4> is norm-preserving, it is clear that the sum in brackets is always finite. Also, ": Dir (G;v) - Dir (Gz) is the composition of the homomorphism 4>* and : Dir (G K ) - Dir (Gz).
CH. 3. §2. ENUMERATION AND ZETA FUNCTIONS IN SPECIAL CASES
67
Now consider the zeta function (llE Dir (G ll). Then cP*{(a) (z)
=
L: ( L:
IEG K
I)I-z
~(M)=I
=
L: a(l)I-Z,
IE6 K
where a(l) denotes the total number of isomorphism classes M such that cP{M)=I. Also, the Euler product formula for (ll and the fact that cP* is a continuous identity-preserving algebra homomorphism implies that cP*{(ll){z) =
II {(I-{P'')-z)-I:
= II n {(I
prime ideals Pin D, r ~ I}
eo
,=\
- p-rz)-l: prime ideals P in D}
where (GK EDir (G K) is the zeta function of GK . . (Recall that (K(Z)=(G K (z).) In studying the enumerative function aD{n), it is sometimes convenient to use the decomposition aD{n) =
2' a(l)
III =n
and first examine a{l). The above formula provides an explicit infinite (-formula for a(l) as a function of I, and shows that a(l) is a PIM-function over GK • In view of the multiplicative properties of aD{n) and a(l), it is of interest to consider the following subcategories ~(p) and lr{P) of lr, where p is any fixed rational prime and P is any fixed prime ideal in D. We define tJ{p) to be the subcategory of all modules of cardinal p" for some n, and we let tJ{P) denote the subcategory of all P-modules in lr (i.e. modules M such that cP{M) is some power of p). If IPI =q, the earlier discussion shows that q is a rational prime-power and that one may regard tJ{P) as an additive arithmetical category with the degree mapping i){M) = log, card (M).
Further, the category tJ{P) then has exactly one prime of degree r, for each r = I, 2, .... Therefore lr (P) is 'arithmetically equivalent' to the category d{p), in the sense that there exists a degree-preserving isomorphism between the associated semigroups; hence ~(P) and d{p) have the same generating function. On the other hand, one may regard the category ~(p) as an additive arithmetical category with the degree mapping i){M) = log, card{M).
68
ENUMERAnON PROBLEMS
CH. 3. §3.
Then the generating function for (j(p) is ee
ZlJ(P)(Y) =
L: ao(p")y"· "=0
Also, if (p)=ple' ... Pmem, where the PI are distinct prime ideals in D, el>O and IPII =pl¥.', then the primes for the category (j(p) separate into m infinite sequences (P lr), ... , (Pmr) with the property that (P lr) = (Xlr (r= 1,2, ... ). Therefore the corresponding Euler product formula is
o
ZlJ(p)(Y) =
=
II [(1 r=l
ya,r) ... (1- yamr)]-l
where
L: p(n)y" n=O ~
Z(y) =
is the generating function for the classical partition function pen). In particular, ao(p") may be regarded as a generalized partition function PK,p(n) depending on the algebraic number field K and the rational prime p. By way of comparison with the above formula, it is interesting to note that the generating function for 6(p) in Example 2.5 may also be written as ec
ZI!(p)(Y) =
II Z(ym2). m=l
(Just as the categories .!II and .!II(p) may be regarded as the initial members of the families of categories (j and (j(p), which are 'parametrized' by the underlying algebraic number field K (or its ring D of algebraic integers), so the categories 6 and 6(p) discussed above may also be viewed as initial cases of analogous arithmetical categories 6 0 and 6 0 (p) 'parametrized' by D (or K) above; for further details, see the author [7].)
§ 3. Special functions and additive arithmetical semigroups It was remarked earlier that the study of specific arithmetical functions, and (-formulae (or similar formulae) for these functions, may often be interpreted as the investigation and solution of special types of enumeration problems regarding an arithmetical semigroup or category. In this section, we note how Theorem 2.6.1 combined with the homomorphism
69
SPECIAL FUNCTIONS
CH. 3. §3.
applies to functions on an additive arithmetical semigroup, and how, for some particular classes of semigroups, this leads to especially detailed and complete arithmetical information. Firstly, let G denote an additive arithmetical semigroup with degree mapping and, for a suitable c>- I, consider the associated norm mapping /al=clJ(G) for aEG. Write ~
a,
ylJ(G) = la!-z = c-IJ(G)z,
so that, for fE Dir (G), fez)
=
Zf(a)!al- Z
GEG
=
Zf(a)yIl(G) =f*(y),
GEG
say. In keeping with our use of the symbols f(a) and fez), which hopefully has not created difficulty thus far, we shall sometimes write
l(q) =
Z f(a),
f* (u) =
IGI=q
so that fez) =
Z !(q)q-Z,
qE IGI
f* (y)
=
Z
f(a),
Z
f* (u)yu.
8(G)=u
uEiI(G)
Thus, for example, ~G(q)=G(q), CG*(u)=G'*'(u), and Zdy)
= CG*(y) = ~dz).
By using Theorem 2.6.1 and the definition of the Mobius function, applying the homomorphism -, and changing to the symbolic powers yU, one now obtains:
3.1. 1beorem. The arithmetical functions defined in §§ 5-6 of Chapter 2 satisfy the following formulae: (i) fl* (y) = IjZdy)· (ii) dk*(y) = [ZG(y)]k; hence d*(y) = [ZG(YW. (iii) d 2* (y) = [Zdy)]4jZG (Y). (iv) d* * (y) = [ZG (yWjZG (y2). (v) p. (y) = ZG(y)ZG(y)ZG(y)jZG(y8). (vi) qk'*' (y) = ZG(y)jZdyk)· (vii) A'*' (y) = ZG(y)jZG(y). (viii) Ak*(Y) = Zdy)jZdy)ZG(yk) if k is even, = Zd y2)Zdyk)jZG(y)ZG(yk) if k is odd. (ix) ~* (y) = (ZGEG NG*(a(a»)ylJ(G»)/ZdY). (x) A * (y) = (log c)yZG' (y)jZG (y), where formal differentiation is defined in the obvious way.
70
ENUMERATION PROBLEMS
CH. 3. §3.
The meaning oif " (ym) for a positive integer m is clear. In the case when G is the semigroup G[q, t] of associate classes of non-zero elements in a Galois polynomial ring GF[q, t], it was shown in Example 2.2 above that Hence Theorem 3.1 implies 3.2. Proposition. The following formulae hold for functions on the additive arithmetical semigroup G[q, t]: (i) (ii) (iii) (iv) (v)
fL'" (y) = 1- qy. dk"'(y) = (I_qy)-k; hence d*(y) = (l_qy)-2. d 2"'(y) = (l_qy)-4(1_ qy2). d*"'(y) = (l_qy)-2(I_ qy2). fJ* (y) = (I - qy6)/{(I_ qy) (I _ qy2)(1 _ qy3)}.
(vi)
= Z::~qnyn+ Z:=dqn_qn-k+l)yn. ..1."'(y) = Z:=oqn y2n- Z:=oqn+l y2n+l. Ak"'(y) = (I-qy)(l- qyk)/(I- qy2) if k is even,
(vii) (viii)
qk*(Y)
= (I-qy)(l- qy2k)/{(I_qy2) (I_qyk)}
(ix)
q>* (y) = 2~:=0 q2n y ll .
(x)
A * (y) = (log c) Z:=o qn+ 1 yn+1.
if k is odd.
Some of the above formulae have quite simple and interesting arithmetical interpretations: For example, formula (vi) shows that every polynomial of degree less than k is k-free (as is obvious directly), but that, for n~k>l, the 'density' of k-free polynomials of degree n in GF[q, t] is
«:»
Next, for any additive arithmetical semigroup G, let (u) and M odd'" (u) denote the total numbers of square-free elements a of degree u such that w(a) is even or odd, respectively. Also, let N eve n '" (u) and N od d '" (u) denote the total numbers of elements a of degree u such that Q(a) is even or odd, respectively. Then
while G'" (u) = N eve n '" (u)
+ N odd *' (u),
J.'" (u) = N eve n '" (u) - N od d '" (u).
CH. 3. §3.
71
SPECIAL FUNCTIONS
In the special case when G is the above semigroup G[q, t], formula (i) above shows that Meven*(I)=O and M o dd " (I)=q (which is again directly obvious), and then that Meven*(n) = Modd"(n)
for n
:» I.
(Although this may seem plausible, it is not an a priori conclusion.) On the other hand, for n >0, formula (vii) and the above equations show that N eve n * (2n) = !(q2n + qn),
N odd*(2n) = !(q2n_ qn),
N even* (2n + I) = !(q2n+l_ qn+l),
N odd* (2n + I) = !(q2n+l+ q,,+I).
In particular it follows that when n is even, when n is odd. We conclude this section with a few remarks on additive arithmetical semigroups of a type that arises frequently in the theory of group representations, and also in the theory of vector bundles in topological K 7theory, for example. These are semigroups for which the degree mappings are integer-valued (and usually stem from 'dimension' functions on categories), and which contain only finitely many primes. Let G denote such an additive arithmetical semigroup. Then the Euler product formula shows that ZG(y) is a rational function of y of the form m
ZG(y) =
II (I-y,,)-I,
;=1
the r; being positive integers. For any function IE Dir (G) that possesses a finite (-formula, the associated power series 1* (y) will then be a rational function of the form
I" (y)
M
=
[J (I
- ySj)-k j,
j=l
where the Sj and k j are rational integers and Sj >0. In any such case, one can in principle obtain complete information about the values I" (n) for n = I, 2, ... , by expanding the above product into a power series. (In some cases, especially if one seeks the asymptotic behaviour of I" (n) as n ...... =, it may help to first decompose I" (y) into partial fractions A (tx- y)-', where r is a positive integer.) Therefore, given sufficient information to
72
ENUMERATION PROBLEMS
CH. 3. §3.
be able to write down the Euler product for ZG(y), one can in principle calculate the numbers G'" (n) and f* (n) for n = 1, 2, ... , when f is any of the functions occurring in Theorem 3.1 for example. If information about the Euler product for G is not available, or only partially available, it is sometimes still possible to draw certain arithmetical conclusions. For example, consider: 3.3. Proposition. Let G denote an additive arithmetical semigroup for which the degree mapping () is integer-valued, and which contains only finitely many primes pEP. Then for all except perhaps a finite number of integers n.
Proof. Under the stated hypotheses, ZaCy) has an expression as a rational function of y of the form noted above. Therefore m
JL* (y)
= II (1- )1"), j~l
where the r/ are certain positive integers. This is a polynomial of degree N=r1+,..+rm • Therefore, for ns-N,
o Selected bibliography for Chapter 3 Section 1: Knopfmacher [9]. Section 2: Bender and Goldman [I], Doubilet, Rota and Stanley [1], Harary [1], Hararyand Palmer [1], Knopfmacher [I, 3, 5-9]. Section 3: Carlitz [1--4], S. D. Cohen [2, 3], Knopfmacher [16], Shader [1, 2].
ENUMERATION PROBLEMS
In many cases, particularly in the study of arithmetical semigroups associated with categories, one may interpret the (-formulae studied in the previous chapter as the answers to certain types of enumeration questions. In the context of arithmetical categories, those questions often also have an intrinsic character essentially independent of the present number-theoretical formulation. A major problem to be considered in this chapter is that of enumerating the isomorphism classes of objects in a given arithmetical category. It will be seen below that the Euler product formula for the zeta function provides a basic stepping-stone towards the solution of this problem, and that it leads to complete algebraic solutions for quite a number of specific categories of interest.
§ 1. A special algebra homomorphism
The reader with some interest in algebra may have noticed quite early that much of our discussion of the Dirichlet algebra Dir (G) of an arithmetical semigroup G would be largely unaffected if complex-valued functions were replaced by functions f: G -k, where k is a general field with a real valuation I 1 (say). This would especially be the case if k were a field of characteristic 0 and one could factorize the norm mapping on G through k and its valuation I I. In this way, one would obtain a generalized Dirichlet algebra Dir (G, k), whose properties might perhaps be worth studying in detail. In this book, it will not be necessary to go further into the above possibility, but instead it will be useful to consider the possibility of assuming slightly less about the initial semigroup G. For present purposes, it will be sufficient to withdraw the requirement of unique factorization and to consider a semigroup H of the following type: Let H denote a commutative semigroup with identity element 1, with the property that every element aEH has only a finite number of divisors in H, and such that there
CH. 3. §1.
A SPECIAL ALGEBRA HOMOMORPHISM
55
exists a real-valued norm mapping I I on H satisfying the conditions: (i) 111=1, and lal>1 for a,cIEH, (ii) labJ = lallbJ for all a, bE H, (iii) the total number NIl (x ) of elements aEH of norm jal.=2x is finite, for each real x e-O. Under these assumptions, let Dir (H) denote the set of all complexvalued functions on H. One may then imitate the earlier treatment of Dir (G) when G is an arithmetical semigroup, and in the first place make Dir (H) into a complex algebra in the obvious similar fashion. Elements of Dir (H) may be represented as formal Dirichlet series over H, and one may define a norm I II on Dir (H) in the same general way as in the discussion of Dir (G). In that case, this 'Dirichlet algebra' for H will retain many of the properties of Dir (G), although it may perhaps not be an integral domain and I I may only be a pseudo-valuation (i.e., have the properties of I II on Dir (G) given in Proposition 2.1.1, except that Ilf(z)g(z)11 = ~ II filII gil without equality necessarily holding in all cases). In particular, these principles may be applied to the semigroup
IGI = 1m {I I: G - R}, where G is some given arithmetical semigroup. Then there is a continuous identity-preserving algebra homomorphism -: Dir (G) - Dir (lGf)
defined by
fez) =
Z ( Z f(a»)q-Z =
qEIG[
[u!=q
~ f(a)
aEG
la\-Z,
given fE Dir (G). (The norm on IGI is taken to be the identity mapping.) It should be noted that the present Dirichlet series over IGI are still formal series, but it is through these that one may make a convenient transition to ordinary series of analysis when desired at a later stage. For the moment, without making such a transition, it is clear that the earliermentioned problem of enumerating the isomorphism classes in an arithmetical category is intimately related with the associated enumerating function 'G(z) = la[-: = G(q)q-Z
Z
aEG
.z
qEIG[
where G(q) denotes the total number of elements aEG of norm lal=q. Now observe that, since - is a continuous identity-preserving algebra homomorphism, it will preserve pseudo-convergent sums and products. (Note that the concept of pseudo-convergence may be carried over to
56
ENUMERATION PROBLEMS
CH. 3. §l.
Dir (H), and so to Dir (IGI), in the obvious way.) Roughly speaking, this remark may be paraphrased so as to state that the homomorphism "preserves formulae". In particular, the Euler product formula for CO
where IGlo= IGj"'{I}, and P(q) is the number of primes of norm q in G. Given a knowledge of the norms Ipl of the primes pEP, this formula provides a mechanical algebraic procedure for calculating the successive numbers G(q) as qE IGI increases monotonically. In other words, this formula, which (if confusion seems unlikely) we may also refer to as the Euler product formula, in principle provides a complete algebraic solution to the enumeration problem discussed above. Before illustrating this statement by means of examples, we now introduce some further terminology and notation, which seems especially appropriate and convenient for certain types of arithmetical semigroups: If one re-examines the examples listed in Chapter I, it will be noticed that in some cases the norm functions are not entirely "natural", in that they arise by exponentiating certain more immediate functions with slightly different properties (e.g. 'dimension' or 'degree' functions). This suggests the formal concept of an additive arithmetical semigroup, which we define to be a semigroup G with the same unique factorization properties as in Chapter I, § I, but with the norm mapping I I now replaced by a realvalued degree mapping 0 on G such that (i) 0(1)=0, o(p»O for pEP, (ii) o(ab)=o(a)+o(b) for all abc G, (iii) the total number N G '" (x) of elements aE G of degree o (a) ~x is finite, for each real x e-O. Obviously, this concept is only formally different from that of an arithmetical semigroup, since one may clearly interchange norm and degree mappings by the rules
o(a) = loge lal
(aE G),
where c> I is fixed. However, as indicated above, it is nevertheless sometimes more natural or convenient to use the new terminology. It is clear, and we shall take as understood, how one may similarly define an additive arithmetical category (i and its associated additive arithmetical semigroup Grt •
CH. 3. §l.
A SPECIAL ALGEBRA HOMOMORPHISM
57
Now let G denote an additive arithmetical semigroup with degree mapping o. Bya remark at the beginning of Chapter I, § I, it is clear that the above conditions (i) to (iii) are equivalent to (i) and (ii) together with (iii)' the total number 1tG * (x) of primes pE P of degree o(p) -;§x is finite, for each x>O. Now consider the enumerating function (dz) for G that arises by associating with G the norm function
\aJ
=
ca(o)
(aEG),
where c> I is fixed by some suitable choice. If confusion seems unlikely, we may also refer to (G(z) as the zeta/unction of G (or of the arithmetical category (£ to which G is associated, if appropriate) and sometimes later even omit the tilde -. In the present situation, we then have (G(z)
=
Z oio«: = Z
qE IGI
G*(u)c- UZ,
uEa(G) .
where c_uz=(CU)_Z and G*(u) denotes the total number of elements aEG such that (a) = u. Further, the Euler product formula (in the new sense) may be written as
o
(G(z)
= II (I-c-a(p)=)-l = II pEP
(I
_c-UZ)-P*(u),
uEa(G)o
where P*(u) denotes the total number of primes pEP such that oCp)=u, and o(G)o=o(G)""{O}. Now write y" = c- UZ = (c")-Z for uE o(G), so that in Dir (iGI) we have y"yV = (CU)-Z(CV)-Z = (c"+V)-Z = y"+v.
Thus yU behaves symbolically like a "u th power", and we may write
say. The Euler product formula then becomes
A frequent simplification. Before the reader becomes too irritated or discouraged by symbolic "u th powers" and formal sums and products defined over obscure subsets IGI or oCG) of the field of real numbers, we hasten to emphasize that in a large proportion of naturally interesting special
58
CH. 3. §1.
ENUMERATION PROBLEMS
cases (such as those stemming from or included in Chapter I) the natural norm or degree functions take only rational integer values. (If the degree function takes only rational integer values, one can always choose e so that the corresponding norm function takes only rational integer values.) In cases of this kind, there is no need to use generalized Dirichlet algebras of the form Dir (H). For, in the first place, if G has an integer-valued norm mapping I I, then one may work directly with the homomorphism -: Dir (G)
given by
fez) =
Dir (G z )
-+
Z ( Z !(a»)n-
n=l
z
lal=n
[jE Dir(G»).
The enumerating function (zeta function) may now be written as
Z G(n)n-
'G(Z) =
Z
n=l
(with the tilde - omitted), and the Euler product formula has the form 'G(Z) =
II (I
_m-Z)-P(m).
m=l
Secondly, if G has an integer-valued degree mapping corresponding integer-valued norm of the form
lal
=
ko(a)
a,
one may use a
for aEG,
where k » I is an integer. If one substitutes y=k- z in Dir (Gz ), one then obtains an 'enumerating power series' or generating function ZG(Y) =
Z G~ (n)yn,
n=O
and the Euler product formula becomes
II (I _ym)-p*(m). oo
ZG(Y) =
m=l
Over here, one may view Y as an indeterminate in the ordinary algebraic sense; for, Y is algebraically independent over the subalgebra C of Dir (G z ) generated by 1, and so the set of all power series enY' (with complex c.) forms a subalgebra of Dir (Gz) which is isomorphic to the formal power series algebra C [[t J] discussed earlier.
Z:=o
CH. 3. §2. ENUMERATION AND ZETA FUNcnONS IN SPECIAL CASES
59
Bearing the convenience of these simplifications in mind, one may wonder how necessary it is to consider norm or degree mappings that are not integer-valued. One reason for considering the more general case is that one may then derive and use a certain "Normalization Principle" in the context of Part II of this book. Another reason lies in the fact that certain analytical techniques for obtaining asymptotic enumeration formulae (discussed in Part III below) seem to be more conveniently phrased in terms of general real-valued norms or degrees. (One specific situation, in which it becomes useful if not essential to consider degree mappings that are not integer-valued, is considered in the author's paper [12]; this covers certain natural arithmetical categories with integer-valued norm mappings, which nevertheless seem most easily dealt with in terms of associated degree mappings that need not necessarily take only integral values.)
§ 2. Enumeration and zeta functions in special cases At this stage it may be interesting to re-examine the main examples of Chapter I, and see what information the above general discussion yields when applied to such special but important cases. 2.1. Example: The Riemann zeta function. For the semigroup Gz of positive integers, IGzl = Gz and - is the identity automorphism of Dir (Gz). The zeta function of Gz is then
and is called the Riemann zeta function. (In using this terminology, and in referring to the other special zeta functions below, one often thinks of (z) and the other functions as functions of a complex variable z (or y, if one is considering ZG(y»), and questions of ordinary convergence then become relevant. In this chapter, however, we shall continue to work only with formal series.) For the Riemann zeta function, we have the classical Euler product formula
(z)
=
n {(I_p-Z)-l: rational primes pl.
2.2. Example: Euclidean domains. For the arithmetical semigroups of associate classes of non-zero elements of the domains Z [V -I] and Z [V2] discussed in Chapter I, the enumerating functions are special cases of the Dedekind zeta function treated below. However, the information on prime
ENUMERATION PROBLEMS
60
CH. 3. §2.
elements in these domains listed in Chapter I, § I, allows us to write down the Euler products explicitly in terms of products over rational primes. Thus, for the Gaussian integers Z [y' -I] we have
II {(I_p-2z)-l: rational primes p=3(mod4)} xII {(l_p-Z)-2: rational primes p= 1 (mod4)}.
CZ[Y- 11(z) = (1-2- Z ) - 1
(Here, and throughout this section, the tilde over CG(z) is omitted.) Similarly, for the domain Z [Y2] one obtains
CZ l Y21(Z) = (1_2- ) - 1 II {(I_p-2Z)-1: rational primes p= ±3 (mod 8)} Z
xII {(I_p-Z)-2:
rational primes p=
± 1 (mod 8)}.
A very simple but nevertheless quite interesting example is provided by the semigroup of associate classes of non-zero elements of a Galois polynomial ring GF[q, t], i.e. the polynomial ring F[t] in an indeterminate t over the finite Galois field F=GF(q) with q elements. This is an additive arithmetical semigroup G=G[q, t] with degree mapping 8(J)=degf It is easy to see that there are exactly qn elements of degree n in G[q, t]. Hence G[q, t] has the generating function ec
ZGlq,tj(Y) =
Z qnyn =
n=o
(I_qy)-l.
Consider the corresponding Euler product formula
II (1- y"')-P*
ZGlq,tj(Y) =
m=1
We shall now show how this formula leads to an explicit equation for P* (m), and hence to the following proposition. 2.3. Proposition. The total number ofmonic irreducible polynomials ofdegree m in GF[q, t] is
r»
(m) = m- 1
Z p.(d)qm1d,
dim
where p. is the Mobius function on the positive integers. Proof. By the Mobius inversion principle, the proposition will follow once we have established the equation qm =
Z dP*(d).
dim
CH. 3. §2. ENUMERATION AND ZETA FUNCTIONS IN SPEGAL CASES
61
One way to do this is to take 'logarithms' on both sides of the Euler product formula and then compare coefficients of the resulting series. In order to do this, we could first use the usual 'logarithmic series' formula to define log (I +h) whenever h is a power series with o (h) >0, where the order o(h) is defined as in Chapter 2, § 4. After verifying that log is continuous relative to the (y)-adic topology for series of the form I +h, and that log {(l
+ hI) (I + h2 ) }
= log (I
+ hI) + log (l + h 2) ,
one could then proceed with the details indicated. However, familiarity with the algebra of differentiation may perhaps make it slightly simpler to use 'logarithmic derivatives' instead: Firstly, if
define its formal derivative by the usual formula ee
f' =
Z ncny"-l.
n=1
One can verify that the rule f -.f' is continuous relative to the (y)-adic topology and is a derivation, i.e., satisfies the usual rules for differentiation. If f is a unit, we define its logarithmic derivative to be DL(f)=f'/f Then D L is a continuous operation relative to the (y)-adic topology (since formal differentiation, inversion and multiplication are continuous), and Ddfg)
=
Ddf)+DL(g)
if f, g are both units. (If desired, one could define and treat logarithms and logarithmic derivatives of arithmetical functions over a general arithmetical semigroup, in a very similar way.) Now take logarithmic derivatives on both sides of the above Euler product formula. Then oo
DL(ZG[q,tj(y») = q(l_qy)-1 =
Z qn+ly",
n=o
while on the other hand the continuity and the additive property of D L give
Z mP* (m)y'"-I(I_ y'")-1 m=1 ~
DdZG[q,tj(Y») =
ee
= ZmP*(m)y'"-I(I+y'"+y2m+ ... ). m=l
62
ENUMERATION PROBLEMS
CH. 3. §2.
By comparing the coefficients of yn-l one then obtains qn=
Z
m., m-l+rm=n-l
mP*(m)=
Z
m{,+l)=n
mP*(m)=ZdP*(d), din
since the preceding sum allows r = O. This proves the proposition.
0
2.4. Example: The Dedekind zeta function. Let GK denote the arithmetical semigroup of all non-zero integral ideals in a given algebraic number field K. The enumerating function for GK is (,.{z)
=
Z IEG
I/I-z =
K
=
L: K(n)nn=l
Z
,
where K(n) denotes the total number of ideals of norm n in G K • The function (K (z) is known as the Dedekind zeta function of K. The facts about prime ideals in GK mentioned in Chapter I show that the Euler product formula for G K has the form
where pal, ... , pak are the norms of the distinct prime ideals PI' ... , Pk in the decomposition (p)=p1e' ... Pkek. (Since (p) has normp[K:Ql, it follows that e1tl1+ ... +ektlk=[K:Q].) By direct considerations, or by means of this formula and Corollary 2.4.2, one notes that K(n) is a multiplicative function of nEG z. 2.5. Example: Finite abelian groups. For the category .91 of all finite abelian groups, the zeta function may be written as C,,(z) =
Z a(n)n-
n=l
Z
,
where a(n) denotes the total number of isomorphism classes of abelian groups of order n. The discussion of primes in .91 given in Chapter I shows that here the Euler product may be written as a pseudo-convergent double product:
II {(I_p-rz)-l: rational =II II(I_p-rz)-l
C~(z) =
eo
,=1 =
=
p
II (rz),
'=1
primes
p, r ~
I}
CH. 3. §2. ENUMERATION AND ZETA FUNCTIONS IN SPECIAL CASES
63
by the Euler product formula for the Riemann zeta function. Again, these formulae or direct considerations show that a(n) is a multiplicative function in Dir (G z ). Further, the last equation provides an explicit infinite '-formula for a(n) as an element of Dir (G z ). Now consider the subcategory d(p) of all finite abelian p-groups, where p is a fixed rational prime. Since p is now fixed, for some purposes it is natural to regard d(p) as an additive arithmetical category, with degree mapping defined by ()(A) = log, card (A). In that case, d (p) has exactly one prime of degree r for each r = I, 2, .... Therefore the Euler product formula implies that d(p) has the generating function Z..,(p)(Y) =
n (I - y)-I =
,=1
Z
n=O
p(n)y",
where pen) =a(pn) is the total number of isomorphism classes of abelian groups of degree n in the above sense. In fact, for n >0, pen) is a function of the positive integer n familiar to number theorists as the partition function, which may be defined arithmetically as the total number of ways of partitioning n into a sum of positive integers (where the order of the summands is disregarded). For example, p(5) = 7 since
5 = 1+4 = 1+1+3 = 1+1+1+2 = 1+1+1+1+1
= 2+3 = 2+2+ I. It will be seen later that the study of additive arithmetical semigroups is often, as in this case, closely related to 'additive' number theory in the classical sense. (For more information on this and the partition function, see for example Hardy and Wright [I).) 2.5. Example: Semisimple finite rings. Let 6 denote the category of all semisimple finite rings. The enumerating function for 6 (or G(!) may be written as '(!(z) =
Z S(n)n-
Z
,
n=1
where Sen) denotes the total number of isomorphism classes of rings of cardinal n in 6. In this case, the primes in the category are the isomorphism classes of the various simple finite rings, and the discussion in Chapter I,
ENUMERAnON PROBLEMS
64
CH. 3. §2.
§ 2, shows that the corresponding Euler product may be expressed as a pseudo-convergent triple product (e(Z) = =
n {(I_p-rmSZ)-l: rational primes p, r ~ I, m ~ I} Il Il (l_p-rmaz)-l
r~l
p
m~l
=
n (rm
2z),
r~l
m;i!;l
by the Euler product formula for the Riemann zeta function. The function S(n) of the positive integer n is multiplicative, and the last equation provides an explicit infinite (-formula for it as an element of Oir (Gz). Now consider the subcategory 6(p) of all semisimple finite p-rings, where p is now a fixed rational prime. As with .9I(p), it is convenient to regard this category as an additive arithmetical category, with degree mapping defined by o(R) = log, card (R). Here the discussion in Chapter I shows that the primes of 6(p) may be arranged in a double sequence (P rm) (r,m= 1,2,... ), where o(Prm ) = rm 2 • Hence the Euler product has the form Ze(p)(Y) =
n (1-
r~l m~l
Z s(n)y", ~
yml)-l
=
n=O
where s(n) = S(p") is the total number of isomorphism classes of rings of degree n in 6(p), and s(n) may be viewed as a natural generalized partition function of the positive integer n. For small n, it is easy to calculate s(n); for example, s(5)=8. (Tables of values for p(n) and s(n) when n;§50 are provided in Appendix 2.) By the discussion in Chapter I, it is not difficult to verify that the subcategory 6 c of all commutative rings in 6 is 'arithmetically equivalent' to the category .91, in the sense indicated at the end of Chapter I. Hence 6 c and .91 have the same enumerating function. Similarly, the subcategory 6 c(p) of all p-rings in 6 c (for a fixed rational prime p) is arithmetically equivalent to the category .9I(p), and has the same generating function. 2.6. Example: Semisimple fioite-dimensiooal algebras. Consider the category 6(F) of all semisimple finite-dimensional associative algebras over a given
CH. 3. §2. ENUMERATION AND ZETA FUNCTIONS IN SPECIAL CASES
65
field F. Three cases were discussed in Chapter I, § 2, and in each of them 6(F) is an additive arithmetical category with the degree function o(A) = dimA.
Case (i): F is an algebraically closed field. In this case there is exactly one prime of degree m 2 for each m= 1,2, .... Therefore 6(F) has the generating function co
Ze(F)(Y) = II (1- y"'")-1 = m=1
ee
Z
p(2) (n)y",
n=O
where p(2 l (n) is the total number of isomorphism classes of n-dimensional algebras in 6(F), and, for n>O, p(2)(n) may be interpreted arithmetically (in the classical sense) as the total number of partitions of n into a sum of squares. Case (ii): F is a real closed field R. Here, by the discussion in Chapter I, the primes split into three sequences (Pm), (Qm) and (R m), where o(Pm) = m 2 ,
(m = 1,2, ... ).
Therefore ~
Ze(Rl(Y) =
II[(I_y"'")(l_y 2mz)(l_y4m")]-1 m=1
say, where SR (n) enumerates the isomorphism classes of n-dimensional algebras in 6 (R). Case (iii): F is a finite Galois field GF(q). The description of the primes for 6(F) in this case given in Chapter I shows that now the category 6(F) is arithmetically equivalent to the category 6(p) above. Hence 6(F) has the same generating function as 6(p) in this case.
2.7. Example: Compact Lie groups, semisimple Lie algebras, and symmetric Riemannian manifolds. The description given under Examples 2.4 and 2.5 in Chapter I shows that the categories considered there may be regarded as additive arithmetical categories. We leave the computation of the generating functions (correct up to a finite number of factors) as an exercise. 2.8. Example: Pseudo-metrizable finite topological spaces. For illustrative purposes, it is interesting to note that the description given in Chapter I shows that the category ~ of these spaces forms an additive arithmetical category,
66
CH. 3. §2.
ENUMERATION PROBLEMS
if one defines the degree of a space X to be card (X). There is then exactly one prime of degree m for each m = I, 2, .... Hence ~ is arithmetically equivalent to the category d(p) above, and has the same generating function p(n)yn, where p(n) is the partition function.
Z:=o
2.9. Example: Finite modules over a ring of algebraic integers. Finally, consider the category ~ = lYD of all modules of finite cardinal over the ring D of all algebraic integers in a given algebraic number field K. If aD(n) denotes the total number of isomorphism classes of modules of cardinal n in ~, then
=
C;v(Z)
Z
n=l
aD(n)n- Z
and the description of the indecomposable modules in shows that
lY given
in Chapter I
II {(1-(I P n- )- l: prime ideals Pin D, r ~ = II II {(I - !PI-rz)-l: prime ideals P in D}
Cil(z) =
Z
I}
=
r~l
where CK(Z) is the Dedekind zeta function discussed above. It follows that aD(n) is a multiplicative function of nE Gz. In studying aD(n) and the category lY or semigroup G;v in general, it is sometimes convenient to factorize the homomorphism - : Dir (G;v) - Dir (Gz) as follows: First note that there is an identity-preserving and norm-preserving homomorphism 4>: G;v -G K defined by sending the isomorphism class of DIP' (P a prime ideal, r s: I) to P', and extending this rule multiplicatively. Then 4> is actually an epimorphism, and 4> induces a continuous identitypreserving algebra homomorphism 4>*: Dir(G,,) - Dir(G K ) according to the rule f
-+
4> * (j), where
4>*(/) (z) =
Z( Z
IEG K
(M)=1
f(M»)/-z.
(Here M denotes the isomorphism class of the module M in lY.) Since 4> is norm-preserving, it is clear that the sum in brackets is always finite. Also, ": Dir (G;v) - Dir (Gz) is the composition of the homomorphism 4>* and : Dir (G K ) - Dir (Gz).
CH. 3. §2. ENUMERATION AND ZETA FUNCTIONS IN SPECIAL CASES
67
Now consider the zeta function (llE Dir (G ll). Then cP*{(a) (z)
=
L: ( L:
IEG K
I)I-z
~(M)=I
=
L: a(l)I-Z,
IE6 K
where a(l) denotes the total number of isomorphism classes M such that cP{M)=I. Also, the Euler product formula for (ll and the fact that cP* is a continuous identity-preserving algebra homomorphism implies that cP*{(ll){z) =
II {(I-{P'')-z)-I:
= II n {(I
prime ideals Pin D, r ~ I}
eo
,=\
- p-rz)-l: prime ideals P in D}
where (GK EDir (G K) is the zeta function of GK . . (Recall that (K(Z)=(G K (z).) In studying the enumerative function aD{n), it is sometimes convenient to use the decomposition aD{n) =
2' a(l)
III =n
and first examine a{l). The above formula provides an explicit infinite (-formula for a(l) as a function of I, and shows that a(l) is a PIM-function over GK • In view of the multiplicative properties of aD{n) and a(l), it is of interest to consider the following subcategories ~(p) and lr{P) of lr, where p is any fixed rational prime and P is any fixed prime ideal in D. We define tJ{p) to be the subcategory of all modules of cardinal p" for some n, and we let tJ{P) denote the subcategory of all P-modules in lr (i.e. modules M such that cP{M) is some power of p). If IPI =q, the earlier discussion shows that q is a rational prime-power and that one may regard tJ{P) as an additive arithmetical category with the degree mapping i){M) = log, card (M).
Further, the category tJ{P) then has exactly one prime of degree r, for each r = I, 2, .... Therefore lr (P) is 'arithmetically equivalent' to the category d{p), in the sense that there exists a degree-preserving isomorphism between the associated semigroups; hence ~(P) and d{p) have the same generating function. On the other hand, one may regard the category ~(p) as an additive arithmetical category with the degree mapping i){M) = log, card{M).
68
ENUMERAnON PROBLEMS
CH. 3. §3.
Then the generating function for (j(p) is ee
ZlJ(P)(Y) =
L: ao(p")y"· "=0
Also, if (p)=ple' ... Pmem, where the PI are distinct prime ideals in D, el>O and IPII =pl¥.', then the primes for the category (j(p) separate into m infinite sequences (P lr), ... , (Pmr) with the property that (P lr) = (Xlr (r= 1,2, ... ). Therefore the corresponding Euler product formula is
o
ZlJ(p)(Y) =
=
II [(1 r=l
ya,r) ... (1- yamr)]-l
where
L: p(n)y" n=O ~
Z(y) =
is the generating function for the classical partition function pen). In particular, ao(p") may be regarded as a generalized partition function PK,p(n) depending on the algebraic number field K and the rational prime p. By way of comparison with the above formula, it is interesting to note that the generating function for 6(p) in Example 2.5 may also be written as ec
ZI!(p)(Y) =
II Z(ym2). m=l
(Just as the categories .!II and .!II(p) may be regarded as the initial members of the families of categories (j and (j(p), which are 'parametrized' by the underlying algebraic number field K (or its ring D of algebraic integers), so the categories 6 and 6(p) discussed above may also be viewed as initial cases of analogous arithmetical categories 6 0 and 6 0 (p) 'parametrized' by D (or K) above; for further details, see the author [7].)
§ 3. Special functions and additive arithmetical semigroups It was remarked earlier that the study of specific arithmetical functions, and (-formulae (or similar formulae) for these functions, may often be interpreted as the investigation and solution of special types of enumeration problems regarding an arithmetical semigroup or category. In this section, we note how Theorem 2.6.1 combined with the homomorphism
69
SPECIAL FUNCTIONS
CH. 3. §3.
applies to functions on an additive arithmetical semigroup, and how, for some particular classes of semigroups, this leads to especially detailed and complete arithmetical information. Firstly, let G denote an additive arithmetical semigroup with degree mapping and, for a suitable c>- I, consider the associated norm mapping /al=clJ(G) for aEG. Write ~
a,
ylJ(G) = la!-z = c-IJ(G)z,
so that, for fE Dir (G), fez)
=
Zf(a)!al- Z
GEG
=
Zf(a)yIl(G) =f*(y),
GEG
say. In keeping with our use of the symbols f(a) and fez), which hopefully has not created difficulty thus far, we shall sometimes write
l(q) =
Z f(a),
f* (u) =
IGI=q
so that fez) =
Z !(q)q-Z,
qE IGI
f* (y)
=
Z
f(a),
Z
f* (u)yu.
8(G)=u
uEiI(G)
Thus, for example, ~G(q)=G(q), CG*(u)=G'*'(u), and Zdy)
= CG*(y) = ~dz).
By using Theorem 2.6.1 and the definition of the Mobius function, applying the homomorphism -, and changing to the symbolic powers yU, one now obtains:
3.1. 1beorem. The arithmetical functions defined in §§ 5-6 of Chapter 2 satisfy the following formulae: (i) fl* (y) = IjZdy)· (ii) dk*(y) = [ZG(y)]k; hence d*(y) = [ZG(YW. (iii) d 2* (y) = [Zdy)]4jZG (Y). (iv) d* * (y) = [ZG (yWjZG (y2). (v) p. (y) = ZG(y)ZG(y)ZG(y)jZG(y8). (vi) qk'*' (y) = ZG(y)jZdyk)· (vii) A'*' (y) = ZG(y)jZG(y). (viii) Ak*(Y) = Zdy)jZdy)ZG(yk) if k is even, = Zd y2)Zdyk)jZG(y)ZG(yk) if k is odd. (ix) ~* (y) = (ZGEG NG*(a(a»)ylJ(G»)/ZdY). (x) A * (y) = (log c)yZG' (y)jZG (y), where formal differentiation is defined in the obvious way.
70
ENUMERATION PROBLEMS
CH. 3. §3.
The meaning oif " (ym) for a positive integer m is clear. In the case when G is the semigroup G[q, t] of associate classes of non-zero elements in a Galois polynomial ring GF[q, t], it was shown in Example 2.2 above that Hence Theorem 3.1 implies 3.2. Proposition. The following formulae hold for functions on the additive arithmetical semigroup G[q, t]: (i) (ii) (iii) (iv) (v)
fL'" (y) = 1- qy. dk"'(y) = (I_qy)-k; hence d*(y) = (l_qy)-2. d 2"'(y) = (l_qy)-4(1_ qy2). d*"'(y) = (l_qy)-2(I_ qy2). fJ* (y) = (I - qy6)/{(I_ qy) (I _ qy2)(1 _ qy3)}.
(vi)
= Z::~qnyn+ Z:=dqn_qn-k+l)yn. ..1."'(y) = Z:=oqn y2n- Z:=oqn+l y2n+l. Ak"'(y) = (I-qy)(l- qyk)/(I- qy2) if k is even,
(vii) (viii)
qk*(Y)
= (I-qy)(l- qy2k)/{(I_qy2) (I_qyk)}
(ix)
q>* (y) = 2~:=0 q2n y ll .
(x)
A * (y) = (log c) Z:=o qn+ 1 yn+1.
if k is odd.
Some of the above formulae have quite simple and interesting arithmetical interpretations: For example, formula (vi) shows that every polynomial of degree less than k is k-free (as is obvious directly), but that, for n~k>l, the 'density' of k-free polynomials of degree n in GF[q, t] is
«:»
Next, for any additive arithmetical semigroup G, let (u) and M odd'" (u) denote the total numbers of square-free elements a of degree u such that w(a) is even or odd, respectively. Also, let N eve n '" (u) and N od d '" (u) denote the total numbers of elements a of degree u such that Q(a) is even or odd, respectively. Then
while G'" (u) = N eve n '" (u)
+ N odd *' (u),
J.'" (u) = N eve n '" (u) - N od d '" (u).
CH. 3. §3.
71
SPECIAL FUNCTIONS
In the special case when G is the above semigroup G[q, t], formula (i) above shows that Meven*(I)=O and M o dd " (I)=q (which is again directly obvious), and then that Meven*(n) = Modd"(n)
for n
:» I.
(Although this may seem plausible, it is not an a priori conclusion.) On the other hand, for n >0, formula (vii) and the above equations show that N eve n * (2n) = !(q2n + qn),
N odd*(2n) = !(q2n_ qn),
N even* (2n + I) = !(q2n+l_ qn+l),
N odd* (2n + I) = !(q2n+l+ q,,+I).
In particular it follows that when n is even, when n is odd. We conclude this section with a few remarks on additive arithmetical semigroups of a type that arises frequently in the theory of group representations, and also in the theory of vector bundles in topological K 7theory, for example. These are semigroups for which the degree mappings are integer-valued (and usually stem from 'dimension' functions on categories), and which contain only finitely many primes. Let G denote such an additive arithmetical semigroup. Then the Euler product formula shows that ZG(y) is a rational function of y of the form m
ZG(y) =
II (I-y,,)-I,
;=1
the r; being positive integers. For any function IE Dir (G) that possesses a finite (-formula, the associated power series 1* (y) will then be a rational function of the form
I" (y)
M
=
[J (I
- ySj)-k j,
j=l
where the Sj and k j are rational integers and Sj >0. In any such case, one can in principle obtain complete information about the values I" (n) for n = I, 2, ... , by expanding the above product into a power series. (In some cases, especially if one seeks the asymptotic behaviour of I" (n) as n ...... =, it may help to first decompose I" (y) into partial fractions A (tx- y)-', where r is a positive integer.) Therefore, given sufficient information to
72
ENUMERATION PROBLEMS
CH. 3. §3.
be able to write down the Euler product for ZG(y), one can in principle calculate the numbers G'" (n) and f* (n) for n = 1, 2, ... , when f is any of the functions occurring in Theorem 3.1 for example. If information about the Euler product for G is not available, or only partially available, it is sometimes still possible to draw certain arithmetical conclusions. For example, consider: 3.3. Proposition. Let G denote an additive arithmetical semigroup for which the degree mapping () is integer-valued, and which contains only finitely many primes pEP. Then for all except perhaps a finite number of integers n.
Proof. Under the stated hypotheses, ZaCy) has an expression as a rational function of y of the form noted above. Therefore m
JL* (y)
= II (1- )1"), j~l
where the r/ are certain positive integers. This is a polynomial of degree N=r1+,..+rm • Therefore, for ns-N,
o Selected bibliography for Chapter 3 Section 1: Knopfmacher [9]. Section 2: Bender and Goldman [I], Doubilet, Rota and Stanley [1], Harary [1], Hararyand Palmer [1], Knopfmacher [I, 3, 5-9]. Section 3: Carlitz [1--4], S. D. Cohen [2, 3], Knopfmacher [16], Shader [1, 2].