Meromorphic continuation of Multivariable Euler product and application

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Meromorphic continuation of Multivariable Euler product and application

Gautami Bhowmik, Driss Essouabri, Ben Lichtin

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Gautami Bhowmik, Driss Essouabri, Ben Lichtin. Meromorphic continuation of Multivariable Euler product and application. 2005. �hal-00004332�

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ccsd-00004332, version 1 - 23 Feb 2005

Meromorphic Continuation of Multivariable Euler Products and Applications.

Gautami Bhowmik

^∗

& Driss Essouabri

^†

& Ben Lichtin

^‡

February 24, 2005

Abstract. This article extends classical one variable results about Euler products defined by integral valued polynomial or analytic functions to several variables. We show there exists a meromorphic continuation up to a presumed natural boundary, and also give a criterion, a la Estermann-Dahlquist, for the existence of a meromorphic extension to Cⁿ. Among applications we deduce analytic properties of height zeta functions for toric varieties over Qand group zeta functions.

Mathematics Subject Classifications: 11M41, 11N37, 14G05, 32D15.

Key words: several variables zeta functions, Euler product, analytic continuation, Manin’s conjecture, Rational points, zeta functions of groups.

Introduction:

There are two fundamental problems in the study of Dirichlet series that admit an Euler product expansion in a region of absolute convergence. The first problem is to prove the existence of a meromorphic continuation into a larger region. Assuming this is possible, the second problem is to describe precisely the boundary of the domain for this meromorphic function. For Dirichlet series in one variable, the first important results go back to Esterman [13] who proved that if h(Y) = P

dF(d)Y^d, where F(d) is a “ganzwertige” polynomial and F(0) = 1, then Z(s) = Q

ph(p^−s) is absolutely convergent for ℜ(s) > 1 and can be meromorphically continued to the half planeℜ(s)>0. Moreover, Z(s) be continued to the whole complex plane if and only ifh(Y) is a cyclotomic polynomial. Subsequently, Dahlquist [6] extended this result tohany analytic function with isolated singularities within the unit circle.

∗Universit´e de Lille 1, UFR de Math., Laboratoire de Math. Paul Painlev´e U.M.R. CNRS 8524, 59655 Villeneuve d’Ascq Cedex, France. Email : bhowmik@math.univ-lille1.fr

†Universit´e de Caen, UFR des Sciences, Campus 2, Laboratoire de Math. Nicolas Oresme, CNRS UMR 6139, Bd. Mal Juin, B.P. 5186, 14032 Caen, France. Email : essoua@math.unicaen.fr

‡Rochester N.Y., USA. Email : lichtin@math.rochester.edu

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The purpose of this paper is to extend these two basic properties to a general class of Dirichlet series that have an absolutely convergent Euler product expansion in some open domain ofCⁿ, n≥2.Thus, the object of our study is an Euler product

Z(h;s) =Y

p

h(p^−s¹, . . . , p^−sⁿ, p)

whenh(X) = 1 +P

kh_k(X₁, . . . , X_n)X_n+1^k is either a polynomial or analytic function with integral coefficients. An essential role in our analysis is played by a polyhedron in Rⁿ, determined by the exponents of monomials appearing in the expression for h(X). This polyhedron plays an important role in the Singularities literature, so it is, perhaps, not too surprising to see it appear here as well.

We first show that there is a meromorphic continuation up to a presumed natural boundary, whose geometry is that of a tube over the boundary of a convex set. Our second main result applies to the case in whichh depends only uponX₁, . . . , X_n.In this event, we prove a very precise result that is the multivariate extension of the work of Estermann-Dahlquist. This shows that the presumed natural boundary isthe natural boundary (in the sense given to this expression in the statement of Theorem 2 in§1.2), unlesshis a “cyclotomic” polynomial.

These results are proved in Section 1.

There are several subjects, such as group theory, algebraic geometry, number theory, knot theory, quantum groups, and combinatorics, in which multivariate zeta functions can arise.

Some of these are discussed in the survey article of [24]. It would therefore be interesting to find applications of our results/methods to the analysis of such zeta functions. We discuss two applications in Sections 2, 3.

The first application (see Section 2) originates with Manin’s conjecture for toric varieties over Q. This gives a precise description of the density of rational points with “exponential height” at mostton such a variety. Solutions to this conjecture have been given by several authors ([1], [8], [22]) (also see [21]). In particular, the method of de la Bret`eche used the deep work of Salberger to meromorphically continue a certain generalized height zeta function into some neighborhood of exactly one point on the boundary of its domain of analyticity. This function was a multivariate Dirichlet series with Euler product in the domain of absolute convergence. His approach sufficed to deduce the density asymptotic of interest for the conjecture, and also gave a strictly smaller order (int) error term. On the other hand, it did not address two general questions. The first inquires about all the other points on the boundary of the domain of analyticity of the Dirichlet series, in particular, how can they be characterized/detected in general, or even calculated in concrete examples. The second asks for an approximation to the natural boundary of the meromorphic extension of the Dirichlet series.

Describing precisely the entire boundary of the domain of analyticity for this series is needed to derive the asymptotic of rational points on the toric variety within a large family of expanding boxes. In the statement of Manin’s conjecture, only one expanding box appears, that with sides all of the same length. There is however, no a priori reason why this expanding box should be privileged over any other. Finding the natural boundary of its meromorphic continuation appears to be an interesting analytic problem by itself, and has

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not, to our knowledge, received any prior attention in the literature. It may even encode something nontrivial about the toric variety. One reason for believing this is the observation that the boundary determines an estimate for the natural boundary of a family of height zeta functions in one complex variable. As such, it offers certain constraints upon the behavior of the zeroes and poles of each height function in this family. Presumably, knowing something about such points, zeroes especially, ought to be interesting.

We solve both of these problems, using the methods developed in Section 1. As a result, our point of view is rather different from that in the works cited above. Our main results are given in Theorems 4, 5, 6 in§2.3. Additional discussion that contrasts our method and results with earlier work can be found in§2.3 following the statement of Theorem 4. The last result in§2 is Theorem 7 in§2.4. This addresses a general (and natural) problem in the multiplicative theory of integers, and is a good illustration of our method. Forany n≥3, we give the explicit asymptotic for the number ofn−fold products of positive integers that equal the n^th power of an integer. The earlier papers ([1], [7], [14], [15]) had found the asymptotic when n = 3. However, nothing comparable for arbitraryn > 3 seems to have been reported before in the literature.

The second application originates in group theory. Several authors have associated a Dirich- let series to certain algebraic or finitely generated (nilpotent) groups in order to study the density of finitely generated subgroups of large index. The algebraic structure of the groups that have been studied in this way enable the series to be written as an absolutely convergent Euler product in one variable, whose factor at the prime p is an explicitly given functionh(p, p^−s).In a series of papers, du Sautoy, Grunewald and others ( [11], [12], [10]) have described with some success the analytic properties of such Euler products.

The evidence produced in these papers leads one to believe that when there is “uniformity”

of the Euler product, there should always exist a meromorphic extension, but that deter- mining the natural boundary is rather difficult in general. On the other hand, the property of uniformity will not be satisfied for many other groups. In this case, only results that are less ambitious in nature should be expected. For example, one can hope to study the boundary of analyticity of the group’s Dirichlet series. It is well known that this series has a real pole on this boundary. A fundamental problem had been to show that this leading pole is rational, and that the series is meromorphic in some halfplane that contains the pole.

The main result of [11] established these two properties for any finitely generated nilpotent group.

Our first observation in Section 3 is that the two main properties proved in [11] can be established in a more elementary fashion, using Theorem 1 (see§1.1) and certain diophantine estimates proved in [ibid.]. Our second observation is that the group zeta functions studied in [12], [10], can be meromorphically extended outside a halfplane of absolute convergence by using the method in Section 1. This is simpler than that used in [10]. We also show that the presumed natural boundary agrees with the one given in [ibid.]. The third observation addresses an analogous problem about the density of subgroups inside finite abelian groups of large order [2]. We indicate by a simple example how nontrivial refinements of standard density results can be found by using multivariate zeta functions and Tauberian theorems.

A third example illustrates another way in which the methods of this paper might eventually

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prove useful, but which we will not address further here. In the study of strings overp−adic fields, one encounters Euler products in several variables. For example, in [5], products of 5-point amplitudes for the “open” strings are considered, where the amplitudes are defined asp−adic integrals

A^p₅(k₁, . . . , k₄) = Z

Q²_p |x|^k¹^k²|y|^k¹^k³|1−x|^k²^k⁴|1−y |^k³^k⁴|x−y|^k²^k³ dxdy.

The product Q

pA^p₅ can be analytically continued. Indeed, our methods can certainly be used to prove this. In so doing, one finds interesting relations to the corresponding real amplitudes.

Notations: For the reader’s convenience, notations that will be used throughout the article are assembled here.

1. N={1,2, . . .}denotes the set of positive integers,N0 =N∪ {0}andpalways denotes a prime.

2. The expression f(λ,y,x) ≪y g(x) uniformly in x∈X and λ∈Λ

means there exists A = A(y) > 0, which depends neither on x nor λ, but could eventually depend on the parameter vectory, such that:

∀x∈X and ∀λ∈Λ |f(λ,y,x)| ≤Ag(x).

When there is no ambiguity we omit the word ‘uniformly’ above.

3. For everyx= (x₁, .., x_n)∈Rⁿ, we setkxk=p

x²₁+..+x²_nresp. |x|=|x1|+..+|xn| to denote the length resp. weight of x. We denote the canonical basis of Rⁿ by (e₁, . . . ,e_n).For every α= (α₁, .., α_n)∈Nⁿ₀,we also setα! =α₁!..α_n!. The standard inner product onRⁿ is denoted h,i.

4. For every s ∈ C, and for every non negative k, we define (^s_k) = s(s−1)..(s−k+1)

k! . For

two complex numbersw andz, we definew^z =e^z^log^w,using the principal branch of the logarithm. We denote a vector in Cⁿ by s = (s₁, . . . , s_n),and write s = σ+iτ, whereσ= (σ₁, . . . , σ_n) and τ = (τ₁, . . . , τ_n) are the real resp. imaginary components of s (i.e. σ_i = ℜ(si) and τ_i = ℑ(si) for each i). We also write hx,si for P

ix_is_i if x∈Rⁿ,s∈Cⁿ.

5. Theunit polydisc P(1) is the set {z= (z₁, . . . , z_n)∈Cⁿ : sup_i=1,...,n|zi|<1}. 6. Given α ∈ Nⁿ₀, we write X^α for the monomial X₁^α¹· · ·X_n^αⁿ. For h(X₁, . . . , X_n) =

P

α∈Nⁿ₀ a_αX^α, the set S(h) := {α : a_α 6= 0} is called the support of h. We also set S^∗(h) := S(h)\ {0}. We denote by E(h) the boundary of the convex hull of S{α+Rⁿ : α∈S^∗(h)}. This polyhedron is called theNewton polyhedron ofh . We denote byExt(h) the set of extremal points ofE(h) (a point of E(h) is extremal if it does not belong to the interior of any closed segment of E(h)). Obviously Ext(h) is a finite subset ofNⁿ₀ \ {0}.

Similarly, if A ⊂ Nⁿ₀ \ {0}, we denote by E(A) the boundary of the convex hull of S{ν+Rⁿ₊|ν ∈A}and call it theNewton polyhedron ofA.Its set of extremal points is denoted byExt(A).

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7. IfA is a subset of Nⁿ₀ \ {0}, we define Aeas follows:

(a) If A is infinite set, then Ae denotes the set of ν ∈ A belonging to at least one compact face ofE(A).

(b) IfA is a finite set, then Ae=A.

In either case, it is clear thatAeis a finite subset of Nⁿ₀ \ {0}.The setAeis called the saturation of A.

8. LetAe^o:={x∈Rⁿ₊ : ∀ν∈A,e hx, νi ≥1}be the dual ofA. Lete ι(A) be the smallest weight of the elements ofAe^o.We will callι(A) the index of A. We define

R(A) :={α ∈Ae^o :|α|=ι(A)}.

For everyα∈R(A), let K(A;α) :={ν ∈Ae : hα, νi= 1}.

1 Analytic properties of multivariate Euler prod- ucts

It will be convenient to split the discussion in two parts. The first main result is Theorem 1.

This constructs a meromorphic extension for a large class of multivariate Euler products that converge absolutely in some product of halfplanes ofCⁿ.The second main result, Theorem 2, extends the classical Estermann-Dahlquist criterion for the existence of a meromorphic extension to all ofCⁿ, n≥2.

1.1 Meromorphic Continuation

The first ingredient is the extension of an Euler product, whosep^thfactor h(p^−s¹, . . . , p^−sⁿ) does not explicitly depend upon p by itself, outside its domain of absolute convergence.

This extends Dahlquist’s theorem [6] to several variables.

The following notations will be used. Let Λ be an open subset of Cⁿ, l₁, . . . , l_r : Λ→ C analytic functions, anda₁, . . . , a_r complex numbers. Define the Euler product

Zl(s) =Zl(s₁, . . . , s_n) =Y

p

1 +

Xr k=1

a_k p^l^k⁽^s⁾

, and for anyδ ∈R, set

W(l;δ) =W(l₁, . . . , l_r;δ) :={s∈Λ :∀i= 1, . . . , r ℜ(l_i(s))> δ}

It is clear that s 7→ Zl(s) converges absolutely and defines a holomorphic function in the domainW(l; 1).

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1.1 Meromorphic Continuation

Lemma 1 (i) The function Zl(s) can be continued into the domain W(l; 0) as follows:

there exists a set {γ(n) : n ∈N^r₀} ⊂ Q[a₀, . . . , a_r] such that for every δ > 0, the function G_δ(s) that is defined (and analytic) in W(l; 1) by the equation

G_δ(s) =Zl(s)· Y

n=(n1,...,nr)∈Nr 0 1≤|n|≤[δ−1]

ζ Xr

j=1

n_jl_j(s) −γ(n)

is actually a bounded holomorphic function in W(l;δ), where it can be expressed as an absolutely convergent Euler product.

(ii) When each a_k∈Z, eachγ(n) ∈Z.In this case, part (i) implies that the equation Zl(s) = Y

n=(n1,...,nr)∈Nr 0 1≤|n|≤[δ−1]

ζ Xr

j=1

n_jl_j(s) γ(n)

G_δ(s) (1)

determines a meromorphic extension of Zl(s) to W(l;δ) for each δ >0.

Remark. As a result, only when each γ(n) is integral does it make sense to speak of a meromorphiccontinuation ofZl(s) beyondW(l; 1).For the sake of simplicity, this function, defined by (1), in which each zeta factor means, of course, its meromorphic extension, is not given a distinct notation.

Even when this is not the case, part (i) shows that an analytic extension of Zl(s) is still possible in simply connected subsets of any W(l;δ), from which the branch (resp. polar) locus of each factor ζ(n·l(s))^−γ(ⁿ⁾ ifγ(n) ∈/ Z (resp. γ(n) ∈Z) has been deleted. For in each such subset, one can use the equation in (i) to express Zl(s) as the product of G_δ(s) with a single valued analytic continuation of each of the zeta factors.

Proof of Lemma 1: It suffices to prove part (i) since the proof of (ii) follows from the construction of theγ(n) in (i).

Let δ ∈ (0,1) be arbitrary. To describe the continuation of Zl(s) into W(l;δ), it will be convenient to work with a somewhat larger class of Euler products defined as follows:

Zl(R_δ;s) =Y

p

1 +

Xr

k=1

a_k

p^l^k⁽^s⁾ +R_δ(p;s)

(2) where for allp,s7→R_δ(p;s) is a holomorphic function on W(l;δ) satisfying

R_δ(p;s) ≪l,δ p⁻² uniformly in p and s ∈ W(l;δ). Evidently, Zl(s) = Zl(R_δ;s) when R_δ(p;s)≡0.

Now let us fix some notations:

1. For each m∈N, set

L_m(l) =L_m(l₁, . . . , l_r) :={n₁l₁+. . .+n_rl_r:n₁+. . .+n_r ≥m};

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2. For each γ₁, . . . , γ_r∈C, set Q0[γ₁, . . . , γ_r] =Z if γ₁, . . . , γ_r ∈Z and Q0[γ₁, . . . , γ_r] =Q[γ1, . . . , γ_r] otherwise;

3. N = [2δ⁻¹] ; 4. L(s) :=Q_r

k=1ζ(l_k(s))^−a^k fors∈W(l; 1).

By elementary computations, we obtain that for anys∈W(l; 1) : L(s) =Y

p

Yr

k=1

1 +

XN v_k=1

a_k v_k

(−1)^v^k

p^v^k^l^k⁽^s⁾ +H_N^k(p;s)

where, ∀k = 1, . . . , r, s 7→ H_N^k(p;s) is a holomorphic function in W(l;δ) and satisfies the condition : H_N^k(p;s)≪N p^−δ(N⁺¹⁾ ≪N p⁻² uniformly in p and s∈W(l;δ). It is also clear thata_k∈N impliesH_N^k = 0 once N > a_k.

Thus, there existf₁, . . . , f_m∈ L2(l) andd₁, . . . , d_m∈Q0[a₁, . . . , a_r] such that : L(s) =Y

p

1−

Xr

k=1

a_k p^l^k⁽^s⁾ +

Xm i=1

di

p^fⁱ⁽^s⁾ +K_N(p;s)

wheres7→KN(p;s) is a holomorphic function in W(l;δ), satisfying the condition K_N(p;s)≪_N p⁻² uniformly inp and s∈W(l, δ).

Now an easy computation shows that for everys∈W(l,1) : Zl(R_δ;s)L(s) = Y

p

1 +

Xr

k=1

a_k

p^l^k⁽^s⁾ +R_δ(p;s)

1− Xr

k=1

a_k p^l^k⁽^s⁾ +

Xm i=1

d_i

p^fⁱ⁽^s⁾ +K_N(p;s)

= Y

p

1 +

Xm i=1

d_i p^fⁱ⁽^s⁾ −

Xr

k1=1

Xr

k2=1

a_k₁a_k₂ p^l^k¹⁽^s^)+l^k²⁽^s⁾ +

Xr

k=1

Xm i=1

a_kd_i

p^l^k⁽^s^)+fⁱ⁽^s⁾ +V_N(p;s)

wheres7→V_N(p;s) is a holomorphic function in W(l;δ),satisfying the bound V_N(p;s) ≪N p⁻² uniformly in p ands∈W(l;δ).

We have thus proved that there exist :

1. g₁, . . . , g_µ ∈ L2(l) and constantsc₁, . . . , c_µ∈Q0[a₁, . . . , a_r]

2. for eachp a holomorphic function s7→R_δ,2(p;s) on W(l;δ),satisfying R_δ,2(p;s)≪δ p⁻² uniformly in p and s∈W(l;δ),

such that for everys∈W(l; 1) we have : Zl(R_δ;s)

Yr k=1

ζ(l_k(s))^−a^k =Y

p

1 +

Xµ

k=1

c_k

p^g^k⁽^s⁾ +R_δ,2(p;s)

. (3)

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Since each g_k ∈ L2(l),it is clear that ℜ(g_k(s))>1 for any s ∈ W(l;¹₂) and k = 1, . . . , µ.

This implies that for anyδ^′> max ¹₂, δ : s7→Y

p

1 +

Xµ

k=1

c_k

p^g^k⁽^s⁾ +R_δ,2(p;s)

is an absolutely convergent Euler product that is holomorphic in the domainW(l;δ^′).

It is now evident how to proceed by induction. LetM = [log₂(N + 1)] + 1∈N. Repeating the above processM times, we conclude that there exist:

1. functionsh₁, . . . , h_q ∈ L1(l) and constantsγ₁, . . . , γ_q∈Q0[a₁, . . . , a_r] 2. functionsu₁, . . . , u_ν ∈ L₂^M(l) and constantsb₁, . . . , b_ν ∈Q0[a₁, . . . , a_r] 3. for eachp, a holomorphic functions7→R_δ,M(p;s) onW(l;δ),satisfying

R_δ,M(p;s)≪δp⁻² uniformly inp and s∈W(l;δ) such that for everys∈W(l; 1) we have :

Zl(R_δ;s) Yq

k=1

ζ(h_k(s))^−γ^k =Y

p

1 +

Xν k=1

b_k

p^u^k⁽^s⁾ +R_δ,M(p;s)

, (4)

andthe right side is absolutely convergent (and holomorphic) onW(l;δ) since 2^−M < δ/2.

We now multiply both sides of (4) byQ

{k:h_k∈L_N+1(l)} ζ(h_k(s))^γ^k and set G_δ(s) :=Zl(R_δ;s)·

Y

hk∈L/ _N+1

ζ(h_k(s))^−γ^k

.

In W(l; 1), G_δ(s) = Q

{k:h_k∈L_N+1(l)} ζ(h_k(s))^γ^k ·Q

p(1 +Pν k=1 bk

p^uk^(s) +R_δ,M(p;s)). The preceding shows that the Euler product on the right is absolutely convergent inW(l;δ).In addition, since h_k ∈ L_N₊₁(l) implies ℜ(h_k(s)) > (N + 1)δ > 2, the product over k also admits an analytic continuation into W(l;δ) as an absolutely convergent Euler product.

Thus,G_δ(s), whose individual factors in its definition are, in general, multivalued outside W(l; 1) (with branch locus the zero or polar divisor of the individual factor), admits an analytic continuation into W(l;δ) as an absolutely convergent Euler product. This proves (i).

Part (ii) follows immediately from the fact that eachγ(n) is integral when eacha_kis integral.

Thus, the equation (1) determines a meromorphic continuation ofZl(s) intoW(l;δ).

This completes the proof of Lemma 1.

Leth₀,· · ·, h_dbe analytic functions on the unit polydiscP(1) inCⁿ,satisfying the property h_k(0) = 0 for each k. Convergence in P(1) should be understood as a normalization condition that can be easily weakened without significant changes to the following discussion.

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Define

h(X₁, . . . , X_n, X_n+1) = 1 + Xd k=0

h_k(X₁, . . . , X_n)X_n+1^k , Z(h;s) =Y

p

h(p^−s¹, . . . , p^−sⁿ, p). Given the power series expansion of eachh_k, h_k(X₁, . . . , X_n) =P

α∈Nⁿ₀ a_α,kX^α, we assume throughout the rest of Section 1 that eacha_α,k ∈Z.

This will suffice for the applications of interest in subsequent sections.

To state the first main result, we will also need the following notations, given the functions h, h₀, . . . , h_d.

For each δ∈R, we set:

V(h;δ) :=Td

k=0{s∈Cⁿ:hα, σi> k+δ ∀α∈Ext(h_k), and σ_i > δ ∀i}, and forδ >0 we set:

1. N =h

2(d+2) δ

i+ 1;

2. Y_N := {(α, k) ∈ Nⁿ₀ ×[0, d] : α ∈ S(h_k) and 1 ≤ |α| ≤ N}, r_N := #Y_N, and N(δ) :={n = (n_α,k)∈N^r₀^N : 1≤ |n| ≤δ⁻¹}.

Theorem 1 There exists A > 0 such that Z(h;s) converges absolutely in V(h;A). In addition, Z(h;s) can be continued into the domain V(h; 0) as a meromorphic function as follows. For any δ >0,there exists {γ(n) :n∈ N(δ)} ⊂Z and G_δ(s),a bounded holomorphic function on V(h;δ),such that the equation

Z(h;s) = Y

n=(nα,k)∈N(δ)

ζ

X

(α,k)∈Y_N

n_α,k(hα,si −k) γ(n)

·G_δ(s), (5) a priori valid in V(h;A), extends to V(h;δ) outside the polar divisor of the product over n∈ N(δ). Moreover G_δ can be expressed as an absolutely convergent Euler product in the domain V(h;δ).

Proof: The idea is to reduce the problem to that studied in Lemma 1. The first needed observation is clear.

Lemma 2 Let k∈ {0, . . . , d}, ands∈Cⁿ be such that σ∈(0,∞)ⁿ. Then

α∈S(hinfk)hα, σi= inf

α∈Ext(hk)hα, σi.

Next, we fixA= 2(1 +d) + 1,and lets∈V(h;A).Evidently, this implieshα, σi ≥A|α|for allα6=0, α∈ ∪^d_k=1S(h_k).Thus for eachk∈[0, d], the convergence of h_k on P(1) implies:

X

α∈S(h_k)

a_α,k p^hα,^s^i−k

≤ X

α∈S(h_k)

|aα,k|

p^hα,σi−k ≤ X

α∈S(h_k)

|aα,k| p^A|α|−k

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≤ X

α∈S(hk)

|aα,k| 2^A|α|/2 · 1

p^A²^−k ≪ 1

pÂ²^−k ≪ 1 pÂ²^−d. By the definition ofA,we have Â₂ −d >1. We conclude that

s7→Z(h;s) =Y

p

h(p^−s¹, . . . , p^−sⁿ, p) =Y

p

1 +

Xd

k=0

X

α∈S(h_k)

is a holomorphic function inV(h;A). For anyδ ∈(0, A) and s∈V(h;δ),it is then easy to expressZ(h;s) in the form of (1) by subtracting off a sufficiently large tail of each h_k that depends uponδ. The details are as follows.

As above, N(=N_δ) = [2(d+ 2)δ⁻¹] + 1.For s∈V(h;δ) note that Xd

k=0

X

α∈S(hk)

|α|≥N

≤

Xd k=0

X

α∈S(hk)

|α|≥N

|aα,k| p^hα,σi−k ≤

Xd k=0

X

α∈S(hk)

|α|≥N

|aα,k| p^δ|α|−k

≤ Xd k=0

1

p^δN² ^−k · X

α∈S(hk)

|a_α,k| 2^δ|α|/2 ≪

Xd k=0

1

p^δN² ^−k ≪ 1 p^δN² ^−d. Since ^δN₂ −d >2, we conclude that Z(h;s) can be rewritten for any s∈V(h;A) as

Z(h;s) =Y

p

1 +

Xd k=0

X

α∈S(hk) 1≤|α|≤N

a_α,k

p^hα,^s^i−k +R_δ(p;s)

,

wheres7→R_δ(p;s) is a bounded holomorphic function in V(h;δ) such that R_δ(p;s)≪δ p⁻² uniformly in p and s∈V(h;δ).

The procedure described in Lemma 1 then applies with the finite set of functions s → hα,si −k, when 1 ≤ |α| ≤ N, 0 ≤ k ≤ d. Thus, for any δ ∈ (0, A), Z(h;s) can be analytically continued as a meromorphic function in V(h;δ), whose precise expression is

given by (5).

Remark: The preceding argument actually shows that Z(h;s) converges absolutely in V(h; 1),even if 1< A.This observation will be needed in the proof of Theorem 3 (see§2.1).

The details justifying this assertion are as follows. For anyδ >0,the preceding discussion has shown thatZ(h;s)|_V_(h;A) can be rewritten as

Z(h;s) =Y

p

1 +

Xd

k=0

X

α∈S(hk) 1≤|α|≤N

a_α,k

p^hα,^s^i−k +R_δ(p;s)

, (6)

whereN =N_δ is defined as above, ands7→ R_δ(p;s) is a bounded holomorphic function in V(h;δ) that satisfies the bound R_δ(p;s)≪_δp⁻² uniformly inp and s∈V(h;δ).

Thus, if δ = 1< A, then s 7→ R1(p;s) is a bounded holomorphic function inV(h; 1) such thatR₁(p;s)≪δ p⁻² while the sum of the finitely many terms indexed by thoseα∈S(h_k)

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1.2 The natural boundary

and |α| ≤N₁ satisfies the property that for any compact subset K ⊂V(h; 1), there exists θ_K>0 such that the sum isO(p^−1−θ^K) uniformly inK.Thus, the product in (6) converges

absolutely inV(h; 1).

A simple extension of Theorem 1 will also be useful for the discussion in Section 3. This enlarges the original domain from which one begins the meromorphic extension ofZ(h;s), by allowing someσ_i to be negative. The case when his a polynomial is the most naturally occuring one, so it will be given below. A simple extension to allow suitable rational factors in Xn+1 can also be made, but this need not be done here. As above, we write h= 1 +P_d

k=0 h_k(X₁, . . . , X_n)X_n+1^k ∈Z[X₁, . . . , X_n, X_n+1],and assumeh_k(0) = 0 for each k.Set:

h_k=X

α6=0

a_α,kX₁^α¹. . . X_n^αⁿ, l= (l_α,k)_(α,k), where l_α,k(s) =hα,si −k iff α∈S(h_k).

For anyδ∈R, set

V^#(h;δ) :=

\d k=0

s∈Cⁿ:hα, σi> k+δ ∀α∈Ext(h_k)

.

The proof of the following assertion is now straightforward.

Corollary 1 s7→Z(h;s) can be continued meromorphically from V^#(h; 1) (where Z(h;s) converges absolutely), into V^#(h;δ).

Proof: Apply the proof of Lemma 1 using the map l, as above. It is clear that for any δ, s ∈ W(l;δ) if and only if s ∈ V^#(h;δ). Thus, the expression for the meromorphic continuation ofZ(h;s) in V^#(h;δ) follows directly from (1).

1.2 The natural boundary

In this subsection we work with a single analytic function h(X) = 1 +P

α6=0a_αX^α.In the setting of Theorem 1, one thinks ofhas the function denoted 1 +h₀.Thus,

V(h;δ) :={s∈Cⁿ:hα, σi> δ ∀α ∈Ext(h) and σ_i > δ ∀i}. The second main result of§1 concerns the Euler product

Z(h;s) =Y

p

h(p^−s¹, . . . , p^−sⁿ) =Y

p

1 +X

α6=0

a_α p^hα,^sⁱ

.

Theorem 1 has shown thatZ(h;s) can be meromorphically continued toV(h; 0) from some domain V(h;A), A > 1, where it converges absolutely as an Euler product. Of interest then are conditions satisfied byhthat implyZ(h;s) can or cannot be extended still further.

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Theorem 2 Assume each a_α ∈Z,and there exist C, D >0 such that for all α∈Nⁿ₀,

|a_α| ≤C(1 +|α|)^D.

Then Z(h;s) can be continued to Cⁿ as a meromorphic function if and only if h is ‘cyclotomic’, i.e. there exists a finite set (m_j)^q_j=1 of elements of Nⁿ₀ \ {0} and a finite set of integers{γj =−γ(m_j)}^q_j=1 such that:

h(X) = Yq

j=1

(1−X^m^j)^γ^j = Yq

j=1

(1−X₁^m^1,j. . . Xn^m^n,j)^γ^j.

In all other cases the boundary∂V(h; 0)is the natural boundary. For purposes of this paper, this expression means that Z(h;s) can not be continued meromorphically into V(h;δ) for any δ <0.

Proof: It is clear that if h is cyclotomic then Z(h;s) has a meromorphic extension to Cⁿ. So, it suffices to prove the converse. To do so, it suffices to assume only thatZ(h;s) admits a meromorphic extension to V(h;δ0) for some δ0 <0.The argument to follow will then show that h must be cyclotomic, from which it follows immediately that Z(h;s) is meromorphically extendible toCⁿ.

We denote the elements ofExt(h) by setting Ext(h) ={α1, . . . , α_q}.

By the proof of Theorem 1, the continuation of Z(h;s) into each V(h;¹_r), r = 1,2, . . . is determined by the following property. There exist A ≥ 1, a sequence {γ(m)}m∈Nⁿ₀ of integers, and a strictly increasing sequence of positive integers {Nr}r such that for each s∈V(h;A) and r ≥1 :

Z(h;s) =

Y

m∈Nn 0 1≤|m|≤Nr

ζ(hm,si)^γ(^m⁾

×G_1/r(s), (7)

whereG_1/r(s) is an absolutely convergent Euler product that is bounded and holomorphic in V(h;¹_r). Thus, the extension of Z(h;s) into V(h;¹_r) is given explicitly as a product of G_1/r with the meromorphic continuation into this domain of each of the finitely many zeta factors in (7).

SetEx :={m∈Nⁿ₀ \ {0}:γ(m)6= 0} and Ex₋:={m∈Nⁿ₀ \ {0}:γ(m)<0}.

We have to distinguish two cases:

Case 1: Ex is infinite

As above, letδ₀ <0 be such thatZ(h;s) has a meromorphic continuation to V(h;δ₀).

Letρ₀ be any fixed (and necessarily nonreal) zero of the Riemann zeta function satisfying ℜ(ρ0) = ¹₂.

Fix β = (β₁, . . . , β_n) ∈ (0,∞)ⁿ such that β₁, . . . , β_n are Q-linearly independent, and set Z_β(t) :=Z(h;tβ).

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For allm∈Ex we settm = _h_m¹_,βi ifγ(m) <0, and tm= _h_m^ρ⁰_,βi ifγ(m)>0.

In addition, choose for eachm∈K,r(m)∈Nsatisfying:

r(m)> 2· |m| ·sup_iβ_i

inf_jhαj, βi and r(m)≥ |m|.

It follows thatN_r(m)≥r(m)≥ |m|.By (7), we have for eachm∈Ex andtβ ∈V(h;_r(¹_m₎) : Z_β(t) =Z(h;tβ) =ζ(thm, βi)^γ(^m⁾

Y

m′∈Nn 0\{m}

1≤|m′|≤Nr(m)

ζ(thm^′, βi)^γ(^m^′⁾

G_1/r(m)(tβ). (8)

From the definition ofr(m), it follows that for each α_j ∈Ext(h):

ℜ(hα_j, tmβi)≥ hα_j, βi

2· hm, βi ≥ hα_j, βi

2· |m| ·sup_iβ_i > 1 r(m). Thus,t7→G_1/r(m)(tβ) is holomorphic in a neighbourhood oft=tm. We now distinguish two subcases:

First subcase: Ex− is infinite

Letm ∈Ex₋, so that tm = _h_m¹_,βi >0.It follows that tm is not a pole of ζ(thm^′, βi)^γ(^m^′⁾ for every m^′ 6= m ∈ Nⁿ₀. This is clear if γ(m^′) > 0 since the only possible pole of this function occurs when t = _h_m¹′,βi,which cannot equal tm because tm = _h_m¹_,βi 6= _h_m¹′,βi. If γ(m^′) < 0, then poles of ζ(thm^′, βi)^γ(^m^′⁾ must be zeroes of ζ(thm^′, βi). A classical fact ([23], pg. 30) tells us that there are no positive zeroes ofζ(s). Thus,tm cannot be a pole ofζ(thm^′, βi)^γ(^m^′⁾.On the other hand, γ(m) <0 implies that tm isa zero of Z_β(t) since

|m| ≤N_r(m).

Furthermore, it is clear that the sequence{t^m}^m∈Ex₋of zeroes ofZ_β(t) converges to 0 when

|m| →+∞.

Now, ifZ(h;s) had a meromorphic continuation toV(h;δ₀),thenZ_β(t) would have to have a meromorphic continuation toU(δ₁) :={t∈C:ℜ(t)> δ₁},where δ₁ = sup_1≤j≤q_hα^δ⁰

j,βi <0.

Thus,Z_β(t) would have to be identically zero, which is impossible because eachG_1/r(s) is an absolutely convergent Euler product in V(h; 1/r), and cannot therefore be identically zero. We conclude that in this subcase,Z(h;s) cannot be meromorphically extended to any V(h;δ) whenδ <0.

Second subcase: Ex₋ is finite

Choosea >0 such thatζ(z)6= 0 for|z| ≤a.

Set

B := 2·(sup_iβ_i)· |ρ0| ·(supm∈Ex₋|m|) a·(inf_iβ_i) >0.

Define Ex₊ := Ex \Ex₋, and fix m ∈ Ex₊ such that |m| ≥ B. Then γ(m) > 0 and tm= _h_m^ρ⁰_,βi ∈C\R.

We then observe the following:

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1. for allm^′ ∈Ex₊ satisfying m^′ 6=m,tm is not a pole of ζ(thm^′, βi)^γ(^m^′⁾ (since the only possible pole of this function is _h_m¹′,βi ∈Rand tm6∈R);

2. for allm^′ ∈Ex₋,tm is not a pole of ζ(thm^′, βi)^γ(^m^′⁾.

( if this were false, thenρ:=tmhm^′, βi would be a zero of ζ(s) satisfying:

|ρ|=|t^m| · hm^′, βi= |ρ0| · hm^′, βi

hm, βi ≤ |ρ0| · |m^′| ·(sup_i β_i)

|m| ·(inf_i β_i) ≤ a·B 2· |m| ≤ a

2, which is impossible);

By (8) and the fact that |m| ≤ N_r(m), we conclude that for each m ∈ Ex+ satisfying

|m| ≥B,tm is a zero ofZ_β(t).Since tm→0 when|m| →+∞,it follows that {tm}_{|m|≥B}

contains a sequence of zeroes of Z_β(t) with accumulation point in U(δ₁) if Z(h;s) could be meromorphically extended toV(h;δ₀).As in the first subcase, this is not possible.

Case 2: Ex is finite SetG(s) := Y

m∈Ex

ζ(hm,si)^−γ(^m⁾

Z(h;s). We will prove that G(s)≡1 .

By choosingr sufficiently large in the equation (7), we deduce that:

1. G(s) is an Euler product of the formG(s) =Q

p

P

α∈Nⁿ₀ mα

p^hα,si

,wherem0 = 1,and there existC, D >0 such that m_α≤C(1 +|α|^D) for allα.

2. G(s) converges absolutely inV(h; 0) =∪_rV(h;¹_r).

Suppose that G(s) 6≡ 1. Then there exists α 6= 0 such that mα 6= 0. Now fix β = (β₁, . . . , β_n)∈(0,∞)ⁿ as in Case 1. It follows that the Euler product

t7→R_β(t) :=G(tβ) =Y

p

X

α∈Nⁿ₀

m_α p^thα,βi

converges absolutely in the halfplane {t∈C:ℜ(t)>0}.

SetS :={α∈Nⁿ₀ :m_α6= 0}.Since hα, βi →+∞ as|α| →+∞, it is clear that there exists ν6=0∈ S such that hν, βi= inf_α6=0∈S hα, βi>0. We fix thisν in the sequel.

LetN =h_8hν,βi

infiβi

i+|ν|+ 1∈N. Then we have for ℜ(t)> _2hν,βi¹ and uniformly inp:

X

|α|≥N+1

m_α p^thα,βi

≪ X

|α|≥N+1

|α|^D pℜ(t)·|α|·(inf_iβi)

≪ X

|α|≥N+1

|α|^D p^ℜ(t)·^|α|² ^·(infⁱ^βⁱ⁾

· 1

p^ℜ(t)·^N+1² ^·(infⁱ^βⁱ⁾

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≪ 1

p^ℜ(t)·^N+1² ^·(infⁱ^βⁱ⁾ X

|α|≥N+1

|α|^D 2^|α|^infⁱ^βⁱ^/4hν,βi

≪ 1

p^ℜ(t)·^N+1² ^·(infⁱ^βⁱ⁾ ≪ 1 p². From this we deduce that

R_β(t) =G(tβ) =Y

p

X

α∈Nn

|α|≤N0

m_α

p^thα,βi +V_N(p;t)

,

where t 7→ V_N(p;t) is a holomorphic function that satisfies the bound V_N(p;t) ≪N p⁻² uniformly in p and all t ∈ C such that ℜ(t) > _2hν,βi¹ . Since this Euler product converges absolutely fort= _hν,βi¹ >0,it follows that

Y

p

1 + X

0<|α|≤N

m_α p^thα,βi

also converges absolutely fort= _hν,βi¹ .However, since|ν| ≤Nit follows thatP

p mν

p^thν,βi

t=1/hν,βi

must also converge, which is not possible. Thus, we conclude that G(s)≡1.

As a result, we must have the following equation for alls∈V(h;A):

Z(h;s) = Y

m∈Ex

ζ(hm,si)^γ(^m⁾= Y

m∈Ex

Y

p

1−p^−h^m^,^sⁱ−γ(m)

= Y

p

Y

m∈Ex

1−p^−h^m^,^sⁱ−γ(m)

=Y

p

h^∗(p^−s¹, . . . , p^−sⁿ),

whereh^∗(X) =h^∗(X1, . . . , Xn) =Q

m∈Ex(1−X^m)^−γ(^m⁾=Q

m∈Ex(1−X₁^m¹. . . X_n^mⁿ)^−γ(^m⁾. Since the Euler product factorization is unique, we conclude that h(X) = h^∗(X), which

completes the proof of Theorem 2.

Whenh = 1 +P

α6=0a_αX₁^α¹. . . X_n^αⁿ ∈Z[X1, . . . , X_n],we also have the analog of Corollary 1, whose notation is used below.

Corollary 2 The Euler product Z(h;s) can be continued from V^#(h; 1) toCⁿ as a meromorphic function if and only if h is ‘cyclotomic’. In all other cases V^#(h; 0) is a natural boundary (that is,Z(h;s)can not be continued meromorphically toV^#(h;δ)for anyδ <0).

Proof: The hard part is to prove the necessity, that is,hmust be cyclotomic if a meromorphic extension to Cⁿ exists. As with Theorem 2, we will show this even if there exists an extension intoV^#(h;δ) for someδ <0.By a permutation of coordinates, one can suppose that:

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{k∈ {1, . . . , n}:∃a∈Ns.t. ae_k∈S^∗(h)}={1, . . . , r}.

If the set is empty, then r= 0.

Assuming the set is nonempty, definec₁, . . . , c_r ∈Nby settingc_k= inf{c >0 :ce_k∈S^∗(h)}, for each k = 1, . . . , r. It is clear that c_ke_k ∈ S^∗(h) for each 1 ≤ k ≤ r. If r = n, then c_ke_k ∈ Ext(S^∗(h)) for all k. Setting, for any δ ∈ R, δ^′ = _max^δ

kck and δ^′′ = _min^δ

kck, this implies thatV(h;δ^′′)⊂V^#(h;δ)⊂V(h;δ^′) ifδ ≥0,whileV(h;δ^′)⊂V^#(h;δ)⊂V(h;δ^′′) if δ <0.The assertion in Corollary 2 therefore follows immediately from the proof of Theorem 2.

Let us then suppose thatr < n. We set h^∗(X₁, . . . , X_n) := h(X₁, . . . , X_n)

Yn

k=r+1

(1−X_k)

=

1 + X

α∈S^∗(h)

aαX^α

X

ε∈{0,1}^n−r

(−1)^|ε|

Yn

k=r+1

X_k^ε^k

(ε= (εr+1, . . . , εn))

= 1 + X

α∈S^∗(h)

a_αX^α− Xn

k=r+1

X_k

+ X

ε∈{0,1}n−r

|ε|≥2

(−1)^|ε|

Yn k=r+1

X_k^ε^k+ X

α∈S^∗(h)

X

ε∈{0,1}n−r

|ε|≥1

(−1)^|ε|a_αX^α Yn k=r+1

X_k^ε^k.

For each k≥r+ 1,setc_k = 1.

It is then clear that c_ke_k ∈ S^∗(h^∗) for all k = 1, . . . , n. Moreover, it follows immediately thatσ_k > _c¹

k for each k≥1 implies:

Z(h;s) Yn k=r+1

ζ(s_k)⁻¹=Z(h^∗;s). (9)

Suppose that there exists δ₀ <0 such that s7→Z(h;s) can be meromorphically continued to V^#(h;δ₀).We set δ₁ = δ₀

2

sup

α∈S^∗(h)

Xn k=1

α_k c_k

−1

<0. It is easy to check (exercise left to reader) that V^#(h^∗;δ₁) ⊂ V^#(h;δ₀). This together with the relation (9) then implies that s 7→ Z(h^∗;s) can be meromorphically continued to V^#(h^∗;δ1). Since there exists, for each k, an integer c_k ≥ 1 such that c_ke_k ∈ S^∗(h^∗), the proof in the case r = n applies, from which it follows that h^∗ is a cyclotomic polynomial. The definition of h^∗ then implies that the polynomialhis also cyclotomic. This completes the proof of Corollary 2.

Remark: Thus, forhas above and not cyclotomic, the position of∂V(h; 0) for a polynomial can differ rather significantly from that for an analytic function. Indeed, for the latter,

∂V(h; 0) is always a union of coordinate hyperplanes, whereas for the former,∂V(h; 0) need notbe a subset of∂[0,∞)ⁿ.The situation is much less clear whenh=h(X1, . . . , Xn, Xn+1) andZ(s) =Q

ph(p^−s¹, . . . , p^−sⁿ, p) is defined as in Theorem 1 (see§3.2).