An Explicit Prime Geodesic Theorem for Discrete Tori and the Hypergeometric Functions

Home , Prime geodesic

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2πia δ value are given by λ = qe X for a = 1, 2,...,δX . As the prime number theorem is obtained via the Riemann zeta function, these formulas are also related to the zeta function ZX (u), called the Ihara zeta function [I], associated with X deﬁned by the following Euler product;

l(P ) −1 −1 ZX (u)= 1 − u (|u|

∞ d (1.3) u log Z (u)= N (n)un, du X X n X=1 one obtains (1.1) by the following determinant expression of ZX (u) with respect to W ;

−1 ZX (u) = det I|E| − uW . Here, for a positive integer m, Im is the identity matrix of size m. Remark that we will encounter another type of the determinant expression of ZX (u) (see § 2.1). The aim of this paper is to establish an explicit prime geodesic theorem, which are not an (d) d asymptotic formula, for the discrete tori. Here, for M = M = (m1,...,md) ∈ (Z≥3) , the (d) d discrete torus DTM of dimension d is deﬁned by the Cayley graph of the group j=1 Z/mjZ d associated with the generating set {±δ1,..., ±δd} with δj = (0,..., 0, 1, 0,..., 0) ∈ Qj=1 Z/mjZ. This is a 2d-regular graph having kMk = m · · · m vertices and dkMk edges. Because of the 1 d Q simplicity of the structure of the graph, their harmonic analysis are well studied. In particular, very recently, there are various results on the number of spanning trees, which is sometimes called a complexity, of the discrete tori and their degenerated ones by establishing the theory of the heat kernel on the graphs [CJK1, CJK2, CJK3, Lo1, Lo2].

To state the result, we need a generalization of the Jacobi polynomial: For α = (α1,...,αd) ∈ n R , β ∈ R and k ∈ Z≥0, deﬁne

(α,β) (|α| + 1)k (d) −k, k + |α| + β + 1 1 − x 1 − x (1.4) Pd,k (x)= FC ; ,..., , k! α1 + 1,...,αd + 1 2 2 !

Γ(a+k) where (a)k = Γ(a) is the Pochhammer symbol with Γ(x) being the gamma function, |α| = α1 + · · · + αd and, for a, b, c1,...,cd ∈ R,

n1 nd (d) a, b (a)|n|(b)|n| x1 xd FC ; x1,...,xd = · · · c ,...,cd (c ) · · · (c ) n ! n ! 1 d 1 n1 d nd 1 d n=(n1,...,nXd)∈(Z≥0) 1 1 (|x1| 2 + · · · + |xd| 2 < 1)

(d) is the Lauricella multivariable hypergeometric function of type C. Notice that, though FC is in (α,β) general an inﬁnite series, Pd,k (x) is actually a polynomial (of degree at most k) in x because of (1) the specialization a = −k. Remark that FC is equal to the Gauss hypergeometric function 2F1 (α,β) (2) and hence P1,k (x) coincides with the classical Jacobi polynomial. When d = 2, FC equals the 2 Appell hypergeometric function F4. It is worth commenting that, if α ∈ (Z≥0) , then the above (α,β) generalized Jacobi polynomial P2,k (x) can be expressed by the generalized hypergeometric function 4F3 (more precisely, see (2.24) in Example 2.8). However, for general d ≥ 3, we can not conﬁrm such degeneracies. Prime geodesic theorem for discrete tori 3

(α,β) We further need a modiﬁcation of special value of Pd,k (x). For 0 ≤ h ≤ n and z = d (z1,...,zd) ∈ (Z≥0) with |z| = h, put

n−h 2n −(2d − 1) 2 h 2d − 3 X(d) (n; z) = 2(d − 1)δ + P (z,−1) , M,h h,0 d, n−h n + h z1,...,zd 2 2d − 1 where δ is the Kronecker delta and h = h! denotes the multinomial coeﬃcient. h,0 z1,...,zd z1!···zd! (d) Write NM (n)= N (d) (n) for short. The following is our main result. DTM (d) d Theorem 1.1. Let M = M = (m1,...,md) ∈ (Z≥3) . For n ≥ 3, it holds that

(d) (d) (d) (1.5) NM (n)= kMk mM (z)XM,h(n; z), h n 0≤ ≤ z R(d) h h≡nX(mod 2) ∈ XM ( )

(d) where, for h ∈ Z≥0, RM (h) is a ﬁnite set deﬁned by

z1 ≥···≥ zd, |z| = z1 + · · · + zd = h, R(d)(h)= z = (z ,...,z ) ∈ (Z )d there exists (y ,...,y ) ∈ Zd such that, as a multiset, M  1 d ≥0 1 d  {z ,...,z } = {m |y |,...,m |y |}  1 d 1 1 d d 

 (d) ( d) (d)  and, for z = (z1,...,zd) ∈ RM (h), mM (z) is a multiplicity of XM,h(n; z) given by

(d) d mM (z)=# (y1,...,yd) ∈ Z | as a multiset, {z1,...,zd} = {m1|y1|,...,md|yd|} . n o We remark that though we can not calculate the right hand side of (1.1) explicitly by hand in general, we can definitely calculate the one of (1.5) because it is a finite sum of (special values (α,β) of) polynomials Pd,k (x), whose coefficients are given concretely. We also remark that above result is easily extended to general discrete tori corresponding to the groups Zd/ΛZd where Λ is an integer square matrix of size d satisfying det Λ 6= 0. In this paper, for simplicity, we only consider such diagonal cases. We give a proof of Theorem 1.1 in Section 2. It is achieved by calculating the additive (d) (d) zeta function ζM (s) associated with DTM in two ways: One is done by using the Ihara zeta function and the other is by the theory of the heat kernel obtained in [CJK1]. In Section 3, we (d) (d) investigate a special case, that is, the normalized discrete torus DTm = DT(m,...,m). We give some numerical examples and observations obtained from the examples. We see that, in this case, our formula (1.5) seems to give a graph analogue of a refinement of the prime geodesic theorem for compact Riemannian manifolds of negative curvature, which counts closed geodesics lying in a fixed homology class. Though there are plenty of papers about the error terms in the prime number theorem and prime geodesic theorems for Riemannian manifolds, there seems to be few results concerning the error terms in (1.2) (see [N] for another kind of prime geodesic theorem with an error term for some special types of graphs). We finally remark that, because our formula (1.5) is explicit, it may yield a description of the error terms of (1.2) in the case of the discrete tori.

2 Additive zeta functions

For a (q + 1)-regular graph X = (V, E), deﬁne 1 ζ (s)= . X λ + s λ∈SpecX (∆X ) 4 Yoshinori YAMASAKI

We call ζX (s) an additive zeta function associated with X. Here, ∆X is the combinatoric Laplacian on X, that is, ∆X = (q + 1)I|V | − A with A being the (vertex) adjacency matrix of 2 X. We sometimes understand that ∆X is a linear operator on the C-vector space L (X)= {f : 2 X → C}, endowed with the inner product (f, g)= x∈V f(x)g(x), f, g ∈ L (X), acting by P (∆X f)(x) = (q + 1)f(x) − f(y). {x,yX}∈E Here we denote {x,y}∈ E by the edge which connects vertices x ∈ V and y ∈ V . The idea for (d) obtaining the main result is to calculate the additive zeta function ζM (s)= ζ (d) (s) associated DTM (d) with DTM in two ways; One is via the Ihara zeta function and the other is via the heat kernel (d) on DTM . First, it is useful to give here the explicit descriptions of the eigenvalues and the corresponding (d) (d) eigenfunctions of the combinatoric Laplacian ∆M of DTM . Let

d (d) VM = Z/mjZ = {(x1,...,xd) | xj ∈ {0, 1,...,mj − 1}, j = 1, 2,...,d} , j Y=1 d (d) ∗ 1 1 mj − 1 (VM ) = Z Z = (v1, . . . , vd) vj ∈ 0, ,..., , j = 1, 2,...,d . mj mj mj j=1 Y .

(d) (d) ∗ (d) (d) ∗ Notice that VM , (VM ) are the sets of all vertices of DTM and its dual (DTM ) (see [CJK3]), (d) (d) ∗ respectively. Moreover, for x = (x1,...,xd) ∈ VM and v = (v1, . . . , vd) ∈ (VM ) , put (x, v) = x1v1 + · · · + xdvd.

(d) ∗ Lemma 2.1. For v = (v1, . . . , vd) ∈ (VM ) , let

d − 1 2πi(x,v) λv = 2d − 2 cos 2πvk , φv(x)= kMk 2 e . k=1 X (d) (d) ∗ Then, we have Spec (∆M )= λv v ∈ (VM ) and see that φv(x) is an orthonormal eigenfunc- 2 (d) tion corresponding to λ , that is, φ (d) forms an orthonormal basis of L (DT ). v v v∈(V ) ∗ M M Proof. See [CJK3].

2.1 Ihara zeta functions

Let X = (V, E)bea(q + 1)-regular graph. It is easy to see that

d (2.1) ζ (s)= log det ∆ + sI . X ds X |V | We now show that, using the determinant expression

2 |E|−|V | 2 (2.2) ZX (u) = (1 − u ) det I|V | − uA + qu I|V | of the Ihara zeta function ZX (u) associated with X, the additive zeta function ζX (s) can be written by the logarithmic derivative of ZX (u). Prime geodesic theorem for discrete tori 5

Lemma 2.2. Let X = (V, E) be a (q + 1)-regular graph. Then, it holds that

us us d (2.3) ζX (s)= |V | 2 + 2 us log ZX (us), 1 − us 1 − qus du where s + q + 1 ± s2 + 2(q + 1)s + (q − 1)2 (2.4) us = . p 2q

Proof. Substituting A = (q + 1)I|V | − ∆X into (2.2) with the relation 2|E| = (q + 1)|V |, we have

2 |V | −1 1 − (q + 1)u + qu 2 q−1 (2.5) det ∆ + I = u(1 − u ) 2 Z (u) . X u |V | X Now the desired formula (2.3) is derived from (2.1) and (2.5) with making a change of variable

1 − (q + 1)u + qu2 (2.6) s = s s , us

2 d us that is, (2.4), together with the relation us = − 2 . ds 1−qus

(d) From this lemma, because DTM is a 2d-regular graph (i.e., q = 2d − 1), one immediately obtains the following proposition.

Proposition 2.3. We have

(d) us us d (d) (2.7) ζM (s)= kMk 2 + 2 us log ZM (us), 1 − us 1 − (2d − 1)us du where s + 2d ± s2 + 4ds + 4(d − 1)2 us = . p2(2d − 1)

Remark 2.4. We notice from (2.6) that

1 + (2d − 1)u2 (2.8) s + 2d = s . us We will encounter this quadratic transformation in § 2.4.

d (d) ∞ (d) n (d) Because us du log ZM (us)= n=1 NM (n)us , what we have to do next is to expand ζM (s) in a series in the variable u . s P 2.2 Heat kernels

We next start from the fact that the additive zeta function ζX (s) can be expressed as the Laplace transform of the theta function θX (t) of X deﬁned by

−λt θX (t)= e . λ∈SpecX (∆X )

(This is why we call ζX (s) an additive zeta function. See [JLG] and [JL] for this terminology). Actually, noticing that Spec (∆X ) ⊂ [0, 2(q + 1)], we have ∞ −st (2.9) ζX (s)= e θX (t)dt (Re (s) > 0). Z0 6 Yoshinori YAMASAKI

From a general theory, we know that θX (t) is essentially given by the heat kernel KX (x,t) : V × R>0 → C on the graph X, which is the unique solution of the heat equation ∂ ∆X + ∂t f(x,t) = 0,  limf(x,t)= δo(x).  t→0 Here, o ∈ V is a ﬁxed base point of X and δo(x) is the Kronecker delta, that is, δo(x)=1if x = o and 0 otherwise. The following is well known.

Lemma 2.5. Let X = (V, E) be a (q + 1)-regular graph with and o ∈ V . Let φλ(x) be an orthonormal eigenfunction of ∆X with respect to λ ∈ Spec (∆X ). Then, we have

−λt (2.10) KX (x,t)= e φλ(o)φλ(x). λ∈SpecX (∆) Proof. See, e.g., [C].

(d) (d) Now, let us calculate KM (x,t) = K (d) (x,t) and θM (t) = θ (d) (t). We notice that, in DTM DTM (d) (d) the case of studying DTM , we always take a base point o = (0,..., 0) ∈ VM . From Lemma 2.1 and (2.10), we have 1 K(d)(x,t)= e−λvte2πi(x,v) M kMk v (d) ∗ ∈(DTXM ) and hence

(d) −λvt (d) (2.11) θM (t)= e = kMkKM (o, t). (d) ∗ v∈(DTXM )

It is shown, for example in [KN] (see also [K]), that the heat kernel KZ(x,t) on Z with o = 0 is given by −2t KZ(x,t)= e Ix(2t) (x ∈ Z), where Ix(t) is the I-Bessel function (or the modiﬁed Bessel function of the ﬁrst kind) having the expansion ∞ 1 t 2n+x (2.12) I (t)= . x n!Γ(n + x + 1) 2 n=0 X d Hence, by the uniqueness of the heat kernel, we see that the heat kernel KZd (x,t) on Z can be d d written as KZd (x,t) = j=1 KZ(xj,t) for x = (x1,...,xd) ∈ Z . Moreover, periodizing this as (d) a function on VM , we haveQ d (d) KM (x,t)= KZd (x + z,t)= KZ(xj + zj,t). z∈Qd m Z (z ,...,z )∈Qd m Z j=1 Xj=1 j 1 d Xj=1 j Y This shows from (2.11) that

d (d) (d) −2dt θM (t)= kMkKM (o, t)= kMk e Izj (2t) (z ,...,z )∈Qd m Z j=1 1 d Xj=1 j Y ∞ d (d) −2dt = kMk mM (z)e Izj (2t). h=0 z z ,...,z R(d) h j=1 X =( 1 Xd)∈ M ( ) Y Prime geodesic theorem for discrete tori 7

d Here, one obtains the last equation above as follows: Let W = Sd ⋉ (Z/2Z) with Sd being the symmetric group of degree d. We see that W naturally acts on Zd and can take the set of all partitions of length less than or equal to d as a set of all representatives of the W -orbits on d (d) d Z . For a partition z of the length l(z) ≤ d, we have mM (z) =#( j=1 mjZ) ∩ W z where W z (d) (d) is the W -orbit of z. Then, for h ∈ Z≥0, we have RM (h) = {z ⊢ hQ| l(z) ≤ d, mM (z) > 0} and, d because j=1 Izj (2t) is W -invariant (notice that Ix(t)= I−x(t)), obtain the desired equation. Now,Q from (2.9), it holds that

∞ (d) (d) (d) ζM (s)= kMk mM (z)Sz (s + 2d) h=0 z R(d) h X ∈ XM ( ) ∞ (d) (d) (2.13) = kMk mM (z)Tz (us), h=0 (d) X z∈RXM (h)

(d) where, for z = (z1,...,zd) ∈ RM (h), we put

∞ d (d) −xt Sz (x)= e Izj (2t)dt (Re (x) > 2d), 0 j Z Y=1 1 + (2d − 1)u2 T (d)(u)= S(d) . z z u Here, we have used the relation (2.8). We will study some properties of these functions in § 2.4.

2.3 Proof of Theorem 1.1

Let us give a proof of the main result.

Proof of Theorem 1.1. From (2.7), we have

2 2 d (d) 1 − (2d − 1)us 1 − (2d − 1)us (d) (2.14) us log ZM (us)= −kMk 2 + ζM (s). du 1 − us us

(d) Therefore, based on the equation (1.3), one can obtain a formula for NM (n) by expanding the right hand side of (2.14) in a series in the variable us. The ﬁrst term on the right hand side of (2.14) can be easily expanded as follows:

2 n (2.15) − kMk + 2kMk(d − 1)us + 2kMk(d − 1) us . n≥4 nX: even

Moreover, since we have from (2.13) together with (2.19), which we will prove in § 2.4,

∞ ∞ k 2d − 3 ζ(d)(s)= kMk −(2d − 1) m(d)(z)C P (z,0) uh+2k+1 M  M z d,k 2d − 1  s h=0 k=0 z∈R(d)(h)  X X  XM    8 Yoshinori YAMASAKI

|z| with Cz = , the second term of (2.14) can be written as z1,...,zd ∞ ∞ k 2d − 3 kMk −(2d − 1) m(d)(z)C P (z,0) uh+2k  M z d,k 2d − 1  s h=0 k=0 z∈R(d)(h)  X X  XM  ∞ ∞    k 2d− 3 + kMk −(2d − 1) m(d)(z)C P (z,0) uh+2k  M z d,k−1 2d − 1  s h=0 k=1 z∈R(d)(h)  X X  XM  ∞   (d) h =kMk  mM (z)Cz us h=0 z R(d) h  X  ∈ XM ( )  ∞ ∞   kh + 2k 2d − 3 + kMk −(2d − 1) m(d)(z)C P (z,−1) uh+2k h + k  M z d,k 2d − 1  s d h=0 k=1 z∈R( )(h)  X X  XM  ∞   (d) n =kMk  mM (z)Cz us n=0  (d)  X z∈RXM (n) 

∞ n−h   2 2n −(2d − 1) (d) (z,−1) 2d − 3 n + kMk  mM (z)CzP n−h  us . n + h d, 2 2d − 1 n=1  0≤h≤n−2 (d)  X  h≡nX(mod 2) z∈RXM (h)     (z,0) 2d−3  Here, in the ﬁrst equality, we have used the equation Pd,0 2d−1 = 1 and, for k ≥ 1, |z| + 2k P (z,0)(x)+ P (z,0) (x)= P (z,−1)(x), d,k d,k−1 |z| + k d,k

n which are easily seen from the deﬁnition. Let us write the coeﬃcient of us of the rightmost (d) (d) hand side of the above formula as, say, C(n). Noticing that RM (0) = {(0,..., 0)} and RM (1) = (d) RM (2) = ∅ because mj ≥ 3 for all j = 1, 2,...,d, we have

2d − 3 C(0) = kMk, C(1) = 0, C(2) = kMk · −2(2d − 1) P (0,−1) = −2kMk(d − 1). d,1 2d − 1 This shows that the second term on the right hand side of (2.14) is expanded as follows:

2 (2.16) kMk− 2kMk(d − 1)us

n−h ∞ 2 2n −(2d − 1) (d) (z,−1) 2d − 3 n + kMk  mM (z)CzP n−h  us n + h d, 2 2d − 1 h n n=3  0≤ ≤ z R(d) h  X  h≡nX(mod 2) ∈ XM ( ) 

 (d)  Combining (2.15) and (2.16), and noticing that mM (0,..., 0) = 1, we obtain the desired formula (1.5). This ends the proof.

F (d) 2.4 A quadratic transformation C We here prove the following proposition, which contains a key formula for our results. Prime geodesic theorem for discrete tori 9

d Proposition 2.6. Let z = (z1,...,zd) ∈ (Z≥0) . (1) We have

C |z| + 1 , |z| + 1 4 4 (2.17) S(d)(x)= z F (d) 2 2 2 ; ,..., z |z|+1 C 2 2 x z1 + 1,...,zd + 1 x x ! ∞ |z| 1 |z| 4 n 1 ( + )n( + 1)n ( 2 ) (2.18) = A(d)(n) 2 2 2 x , x|z|+1 z (h + 1) n! n n X=0 |z| where Cz = and z1,...,zd (d) n n + |z| Az (n)= . n1,...,nd n1 + z1,...,nd + zd n1,...,nd≥0 n1+···X+nd=n

(2) We have

∞ 2d − 3 k (2.19) T (d)(u)= C u|z|+1 P (z,0) −(2d − 1)u2 , z z d,k 2d − 1 k=0 X (α,β) where Pd,k (x) is the generalization of the Jacobi polynomial deﬁned by (1.4). Moreover, if d α ∈ (Z≥0) , then it can be written as

k 1 (|α| + 1) (−k) (k + |α| + β + 1) ( 1−x )n (2.20) P (α,β)(x)= k A(d)(n) n n 2 . d,k C k! α (|α| + 1) n! α n n X=0 Proof. Put h = |z|. Using (2.12), we have

1 ∞ dt S(d)(x)= e−xtth+2|n|+1 z n ! · · · n !(n + z )! · · · (n + z )! t d 1 d 1 1 d d 0 n=(n1,...,nXd)∈(Z≥0) Z 1 (h + 2|n|)! 1 = xh+1 n ! · · · n !(n + z )! · · · (n + z )! 2|n| d 1 d 1 1 d d x n=(n1,...,nXd)∈(Z≥0) h + 1 h + 1 4 |n| 1 h! 2 2 |n| 2 |n| ( 2 ) = x . xh+1 z ! · · · z ! (z + 1) · · · (z + 1) n ! · · · n ! 1 d d 1 n1 d nd 1 d n=(n1,...,nXd)∈(Z≥0) 2n a 1 a Here, we have employed the identities (a+n)! = a!(a+1)n and (a+2n)! = a!2 2 + 2 n 2 +1 n for a, n ∈ Z≥0. Hence we obtain (2.17). The formula (2.18) is easily obtained from (2.17). (d) We next concentrate on Gz (u). From (2.17), it holds that

h + 1 h + 1 2|n| 2|n| (d) h+1 2 2 |n| 2 |n| 2 u 2 −(h+1+2|n|) Tz (u)= Czu 1 + (2d − 1)u . (z1 + 1)n · · · (zd + 1)n n1! · · · nd! n=(n ,...,n )∈(Z )d 1 d 1 Xd ≥0 The generalized binomial theorem yields

∞ h n h + l + 2|n| l 1 + (2d − 1)u2 −( +1+2| |) = −(2d − 1)u2 l l=0 X∞ (h + 1)l+|n|(l + |n| + h + 1)|n| l = −(2d − 1)u2 2|n| h 1 h l=0 l!2 2 + 2 |n| 2 + 1 |n| X 10 Yoshinori YAMASAKI and hence

(d) Tz (u)

∞ 2(l+|n|) h+1 (h + 1)l+|n|(l + |n| + h + 1)|n| u l = Czu −(2d − 1)  (z1 + 1)n · · · (zd + 1)n l!n1! · · · nd! l=0 n=(n ,...,n )∈(Z )d 1 d X  1 Xd ≥0 

∞  2k  h+1 (h + 1)k(k + h + 1)|n| u k−|n| = Czu   −(2d − 1) (z1 + 1)n1 · · · (zd + 1)nd (|n|− k)!n1! · · · nd! k  n n ,...,n Z d  =0  =( 1 d)∈( ≥0)  X |Xn|≤k   ∞  1 |n|  h+1 (h + 1)k (−k)|n|(k + h + 1)|n| 2d−1 2 k = Czu   −(2d − 1)u . k! (z1 + 1)n1 · · · (zd + 1)nd n1! · · · nd! k  n n ,...,n Z d  =0  =( 1 d)∈( ≥0)  X |Xn|≤k   Therefore we obtain (2.19). The equation (2.20) follows in the same manner as (2.18).

Example 2.7. When d = 1, we have respectively from (2.17) and (2.19)

h 1 h (1) 1 2 + 2 , 2 + 1 4 Sh (x)= h+1 2F1 ; 2 , x h + 1 x ! ∞ uh+1 (2.21) T (1)(u)= uh+1 P (h,0)(−1)(−u2)k = . h 1,k 1+ u2 n=0 X Here, in the last equality in (2.21), we have used the well-known formula

k + β (2.22) P (α,β)(−1) = (−1)k . 1,k k We remark that (2.21) is also obtained from the Pfaﬀ transformation

a a+1 2 , 2 4x a a, b 2F1 ; 2 = (1+ x) 2F1 ; x a − b + 1 (1 + x) ! a − b + 1 ! for the Gauss hypergeometric functions 2F1. From this, we can say that (2.19) is a kind of (d) generalization of the Pfaﬀ transformation for FC with special parameters.

2 Example 2.8. We next consider the case d = 2. Let z = (z1, z2) ∈ (Z≥0) . It is easy to see that 2n + |z| (2.23) A(2)(n)= . z n + z ,n + z 1 2 Hence, we have respectively from (2.17) and (2.19)

C |z| + 1 , |z| + 1 , |z| + 1, |z| + 1 16 S(2)(x)= z F 2 2 2 2 2 2 ; , z |z|+1 4 3 2 x |z| + 1, z1 + 1, z2 + 1 x ! ∞ 1 T (2)(u)= C u|z|+1 P (z,0) (−3u2)k. z z 2,k 3 Xk=0 Prime geodesic theorem for discrete tori 11

Here, pFq is the generalized hypergeometric function deﬁned by

∞ n a1, . . . , ap (a1)n · · · (ap)n x pFq ; x = . b1, . . . , bq (b ) · · · (b ) n! n 1 n q n X=0 2 Moreover, from (2.20) and (2.23), we have for α = (α1, α2) ∈ (Z≥0)

|α| 1 |α| (α,β) (|α| + 1)k −k, k + |α| + β + 1, 2 + 2 , 2 + 1 (2.24) P2,k (x)= 4F3 ; 2(1 − x) . k! |α| + 1, α1 + 1, α2 + 1 !

3 Normalized discrete tori

(d) 3.1 Prime geodesic theorem for DTm (d) (d) In this section, we consider a special case, that is, a normalized discrete torus DTm = DT(m,...,m) for m ≥ 3. In this case, the result obtained in the previous section becomes more simple form (d) (d) as below. Here, we put Nm (n)= N(m,...,m)(n). Theorem 3.1. For n ≥ 3, it holds that

(d) d (d) (3.1) Nm (n)= m m(µ)Xm,h(n; µ), n µ⊢h 0≤h≤ m mh≡Xn (mod 2) l(Xµ)≤d

m1(µ) m (µ) where, for a partition µ = (µ1,...,µl) = (1 · · · h h ) ⊢ h of length l(µ) = l ≤ d with mj(µ) being the multiplicity of j in µ, d l (3.2) m(µ) = 2l u(µ) with u(µ)= , l m (µ),...,m (µ) 1 h n−mh 2n −(2d − 1) 2 mh 2d − 3 X(d) (n; µ) = 2(d − 1)δ + P (mµ,−1) . m,h h,0 d, n−mh n + mh mµ1,...,mµl 2 2d − 1

Example 3.2. Consider the case d = 1. Let us check the trivial result

(1) 0 m ∤ n, Nm (n)= (2m m | n from our formula. First, we have m(µ)=1 if µ = 0 (the empty partition) and 2 otherwise. Moreover, from (2.22), it holds that

n (1) 1 m | n and h = m , Xm,h(n; µ)= (0 otherwise. Hence, from (3.1), one can actually obtain the desired formula. Example 3.3. We next consider the case d = 2. It holds that

(2) 2 (2) Nm (n)= m m(µ)Xm,h(n; µ) n µ⊢h 0≤h≤ m mh≡Xn (mod 2) l(Xµ)≤2

2 e (2) (2) (2) = m δnXm,0(n; 0) + 4 Xm,h n; (h) + u(µ)Xm,h(n; µ) . n µ⊢h 1≤h≤ m    mh≡Xn (mod 2) l(Xµ)=2        12 Yoshinori YAMASAKI

e Here, δn = 1 if n is even and 0 otherwise. Notice that, for µ = (µ1,µ2) ⊢ h with l(µ) = 2, u(µ)=1if µ1 = µ2 and 2 otherwise. Using (2.20), we have

n n 1 (2) n − 2 , 2 , 2 4 Xm,0(n; 0) = 2 + 2(−3) 2 3F2 ; . 1, 1 3!

n−mh Moreover, for h ≥ 1 and µ = (µ1,µ2) ⊢ h with l(µ) = 2, letting k = 2 , we have

k mh 1 mh (2) 4n(−3) (mh + 1)k −k, k + mh, 2 + 2 , 2 + 1 4 Xm,h(n; (h)) = 4F3 ; , n + mh k! mh + 1, mh + 1, 1 3! k mh 1 mh (2) 4n(−3) mh (mh + 1)k −k, k + mh, 2 + 2 , 2 + 1 4 Xm,h n; (µ1,µ2) = 4F3 ; . n + mh mµ1,mµ2 k! mµ + 1,mµ + 1, mh + 1 3 1 2 ! For example, let us consider the case m = 3 and n = 6. Since

(2) (2) (2) (2) N3 (6) = 9 X3,0 (6; 0) + 4X3,2 6; (2) + 4X3,2 6; (1, 1) with

1 (2) −3, 3, 2 4 11 X3,0 (6; 0) = 2 + 2(−27)3F2 ; = 2 + 2(−27) − = 24, 1, 1 3! 27 7 (2) 0, 6, 2 , 4 4 X3,2 6; (2) = 4F3 ; = 1, 7, 7, 1 3! 7 (2) 6 0, 6, 2 , 4 4 X 6; (1, 1) = 4F3 ; = 20, 3,2 3 4, 4, 7 3 ! we have (2) N3 (6) = 9 1 · 24 + 4 · 1 + 4 · 20 = 9 · 108 = 972. (2) For the other values of X3,h(n; µ) with n ≤ 10, see the table below.

N (2) n 3 ( ) n h =0 h =1 h =2 h =3 32 3 X(1) = 1 4 4 X(0) = 8 8 5 X(1) = 10 40 6 X(0) = 24 X(2) = 1, X(12)=20 108 7 X(1) = 42 168 8 X(0) = 216 X(2) = 40, X(12)=80 696 9 X(1) = 414 X(3) = 1, X(21) = 84 2332 10 X(0) = 1520 X(2) = 420, X(12) = 840 6560

(2) Table 1: The values of X(µ)= X3,h(n; µ)

3.2 Some observations

From the table above, we ﬁrst expect the following. Prime geodesic theorem for discrete tori 13

(d) Conjecture 3.4. It holds that Xm,h(n; µ) ∈ Z≥0.

(d) If Conjecture 3.4 is true, then we can expect that the summand Xm,h(n; µ) in the righthand (d) side of (3.1) itself counts something special type of cycles in DTm , as we will see below: Let (d) (d) us subdivide Nm (n) into small pieces by the following manner. Fix o ∈ Vm . We notice that (d) there is an one-to-one correspondence between a cycle C in DTm starting from and ending to o d of length n and a path C in Z starting from the origin (0,..., 0) and ending to (mp1,...,mpd) d d for some (p1,...,pd) ∈ Z of length n. Let us call a path C in Z reduced modulo m if the (d) (d) corresponding cycle C in DTm is reduced. Let RCm (n) be the set of all reduced cycles in (d) d (d) DTm starting from and ending to o of length n and, for p = (p1,...,pd) ∈ Z , RPm (n; p) the set of all reduced paths modulo m in Zd starting from the origin (0,..., 0) and ending to (mp1,...,mpd) of length n. It is clear that

(d) d (d) Nm (n)= m #RCm (n) d (d) = m #RPm n; p p∈Zd X d (d) (3.3) = m m(µ)Nm,h(n; µ), n µ⊢h 0≤h≤ m mh≡Xn (mod 2) l(Xµ)≤d

(d) where, for µ = (µ1,...,µd) ⊢ h of length l(µ)= l ≤ d, m(µ) is deﬁned in (3.2) and Nm,h(n; µ) is the number of all reduced paths modulo m in Zd starting from the origin (0,..., 0) and ending d to (mµ1,...,mµd) of length n. Notice that m represents the number of choices of the starting points. Now, it is natural from (3.1) and (3.3) to expect the following.

(d) (d) Conjecture 3.5. It holds that Xm,h(n; µ)= Nm,h(n; µ).

(d) It is clear that Conjecture 3.4 follows from Conjecture 3.5 because Nm,h(n; µ) ∈ Z≥0.

Example 3.6. The following ﬁgures 1, 2 and 3 support that Conjecture 3.5 is true for the case (2) (2) (2) (2) d = 2, m = 3 and n = 6, that is, N3,0 (6; (0)) = X3,0 (6; (0)) = 24, N3,2 (6; (2)) = X3,2 (6; (2)) = 1 (2) (2) and N3,2 (6; (1, 1)) = X3,2 (6; (1, 1)) = 20. Here, the black dots in the ﬁgures represent the lattices points. In particular, the big black dots denote points of the form of (mp1,mp2) for some 2 (d) (p1,p2) ∈ Z . By the same manner, we have already checked that the equation Xm,h(n; µ) = (d) Nm,h(n; µ) holds for n ≤ 10.

Let M be a compact Riemannian manifold of negative curvature. It is known that there exist countably infinitely many closed geodesics in M and that a closed geodesic in M corresponds to a unique non-trivial conjugacy class Conj(γ) of γ ∈ π1(M) where π1(M) is the fundamental group of M. Let us write the closed geodesic corresponding to Conj(γ) as Cγ. Let NM (x) be the number of all closed geodesics in M of length ≤ x. Then, we have the following prime geodesic theorem for M ([Mar]); ehx N (x) ∼ (x →∞), M hx where h> 0 is the topological entropy of the geodesic flow over M. This is an analogue of the classical prime number theorem. Moreover, we have also an analogue of the Dirichlet theorem on arithmetic progressions, which counts the closed geodesics lying in a fixed homology class: Let ab H1(M, Z) be the first homology group over Z of M and φ : π1(M) → H1(M, Z) = π1(M) = 14 Yoshinori YAMASAKI

(2) 2 Figure 1: N3,0 (6; (0)) = 24; it is the number of all reduced paths C modulo 3 in Z starting from (0, 0) and ending to (0, 0) of length 6.

(2) 2 Figure 2: N3,2 (6; (2)) = 1; it is the number of all reduced paths C modulo 3 in Z starting from (0, 0) and ending to (6, 0) of length 6.

(2) 2 Figure 3: N3,2 (6; (1, 1)) = 20; it is the number of all reduced paths C modulo 3 in Z starting from (0, 0) and ending to (3, 3) of length 6.

π1(M)/[π1(M),π1(M)] be the natural projection. Here, [π1(M),π1(M)] is the commutant sub- group of π1(M). For a ﬁxed α ∈ H1(M, Z), let NM (x; α) be the number of all closed geodesics C = Cγ in M of length ≤ x satisfying φ(γ) = α. Then, it is shown in [AS, PS, La] that there Prime geodesic theorem for discrete tori 15 exists a constant C > 0, not depending on α, such that ehx (3.4) NM (x; α) ∼ C (x →∞). b +1 x 2

Here, b ∈ Z≥0 is the ﬁrst Betti number of M, that is, the rank of H1(M, Z). Now, one may regard the claim in Conjecture 3.5 as a graph analogue of (3.4) because the (d) partition µ ⊢ h of length at most d appeared in Xm,h(n; µ) can be regarded as an element of d (d) (d) (d) Z ≃ H1(RT , Z) (≃ π1(RT )) with RT being the real torus of dimension d. Let us see this (d) in the situation of Example 3.6. Write H1(RT , Z) = Zα + Zβ with the standard basis α, β respectively corresponding to the meridian and longitude of the torus. Then, the conjecture says (2) (2) (2) (2) that X3,0 (6; (0)), X3,2 (6; (2)) and X3,2 (6; (1, 1)) count cycles in DT3 lying in homology classes 0 (the null homology class), 2α (or 2β by the symmetry) and α + β, respectively.

Acknowledgment

The author would like to thank Professor Hiroyuki Ochiai for carefully reading the manuscript and giving many helpful comments. Moreover, the author also thanks the referee for several useful and variable comments which improve the paper.

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Yoshinori YAMASAKI Graduate School of Science and Engineering, Ehime University, Bunkyo-cho, Matsuyama, 790-8577 JAPAN. [email protected]