On Universality of Some Beurling Zeta-Functions

: Let P be the set of generalized prime numbers, and ζ P ( s ) , s = σ + it , denote the Beurling zeta-function associated with P . In the paper, we consider the approximation of analytic functions by using shifts ζ P ( s + i τ ) , τ ∈ R . We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set { log p : p ∈ P} , and the existence of a bounded mean square for ζ P ( s ) . Under the above hypotheses, we obtain the universality of the function ζ P ( s ) . This means that the set of shifts ζ P ( s + i τ ) approximating a given analytic function defined on a certain strip (cid:98) σ < σ < 1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied.


Introduction
A positive integer q > 1 is called prime if it has only two divisors, q and 1.Thus, 2, 3, 5, 7, 11, . . .are prime numbers.Integer numbers k > 1 that have divisors different from k and 1 are called composite.It is well known that the set of all primes is infinite, and this was first proved by Euclid.By the fundamental theorem of arithmetic, every integer k > 1 has a unique representation as a product of prime numbers.Thus, and q j is the jth prime number, j = 1, . . ., r, with some r ∈ N.
Investigations of the number of prime numbers were more complicated.We recall that a = O(b), a ∈ C, b > 0, means that there exists a constant c > 0 such that |a| ⩽ cb.Comparatively recently, in 1896 Hadamard [1] and de la Vallée-Poussin [2] proved independently the asymptotic formula For this, they applied the Riemann idea [3] of using the function now called the Riemann zeta-function.The distribution low of prime numbers was found.
Prime numbers have generalizations.The system P of real numbers 1 < p 1 ⩽ p 2 ⩽ • • • ⩽ p n ⩽ • • • such that lim n→∞ p n = ∞ are called generalized prime numbers.Generalized prime numbers were introduced by Beurling in [4], and are studied by many authors.The system P generates the associated system N P of generalized integers consisting of finite products of the form r , α j ∈ N 0 , j = 1, . . .r, with some r ∈ N.
The main problem in the theory of generalized primes is the asymptotic behavior of the function π P (x) The function π P (x) is closely connected to the number of generalized integers In these definitions, the sums are taking counting multiplicities of p and m.Distribution results for generalized numbers were obtained by Beurling [4], Borel [5], Diamond [6][7][8], Malvin [9], Nyman [10], Ryavec [11], Hilberdink and Lapidus [12], Stankus [13], Zhang [14], and others.The important place in generalized number theory is devoted to making relations between N P (x) and π P (x).We mention some of them.From a general Landau's theorem for prime ideals [15], we have the estimate that implies Nyman proved [10] that the estimates and with arbitrary α > 0 and α 1 > 0 are equivalent.Beurling observed [4] that the relation is implied by (2) with α > 3/2.
It is important to stress that Beurling began to use zeta-functions for investigations of the function π P (x).These zeta-functions ζ P (s), now called Beurling zeta-functions, are defined in some half-plane σ > σ 0 , by the Euler product or by the Dirichlet series where σ 0 depends on the system P.
Suppose that (1) is true.Then, the partial summation shows that the series for ζ P (s) is absolutely convergent for σ > 1, the function ζ P (s) is analytic for σ > 1, and the equality This gives analytic continuation for ζ P (s) to the half-plane σ > β, except for the point s = 1 which is a simple pole with residue a.
Beurling zeta-functions are attractive analytic objects; investigations of their properties lead to interesting results, and require new methods.Various authors put much effort into showing that the Beurling zeta-functions have similar properties to classical ones.We mention a recent paper [16] containing deep zero-distribution results for ζ P (s).
In this paper, we investigate the analytic properties of the function ζ P (s).The approximation of analytic functions is one of the most important chapters of function theory.It is well known that the Riemann zeta-function ζ(s) is universal in the sense of approximation of analytic functions.More precisely, this means that every non-vanishing analytic function defined on the strip {s ∈ C : 1/2 < σ < 1} can be approximated with desired accuracy by using shifts ζ(s + iτ), τ ∈ R. Universality of ζ(s) and other zeta-functions has deep theoretical (zero-distribution, functional independence, set denseness, moment problem, . . . ) and practical (approximation problem, quantum mechanics) applications.On the other hand, the universality theory of zeta-functions has some interior problems (effectivization, description of a class of universal functions, Linnik-Ibragimov conjecture, see Section 1.6 of [17], . . .); therefore, investigations of universality are continued, see [17][18][19][20][21][22][23].
Our purpose is to prove the universality of the function ζ P (s) with a certain system P.We began studying the approximation of analytic functions by shifts ζ P (s + iτ) in [24].Suppose that the estimate (1) is valid.Let Suppose that σ < 1 and define Here, and in the sequel, the notation a ≪ c b, a ∈ C, b > 0, shows that there exists a constant c = c(ε) > 0 such that |a| ⩽ cb.Denote by H(D) the space of analytic on D functions equipped with the topology of uniform convergence on compacta, and by measA the Lebesgue measure of a measurable set A ⊂ R. The main result of [24] is the following theorem.
Theorem 1. Suppose that the system P satisfies the axiom (1).Then there exists a closed non-empty subset F P ⊂ H(D) such that, for every compact set K ⊂ D, f (s) ∈ F P and ε > 0, lim inf Moreover, the limit exists and is positive for all but at most countably many ε > 0.
Theorem 1 demonstrates good approximation properties of the function ζ P (s); however, the set F P of approximated functions is not explicitly given.The aim of this paper, using certain additional information on system P, is to identify the set F P .
A new approach for analytic continuation of the function ζ P (s) involving the generalized von Mangoldt function and was proposed in [12].Let, for α ∈ [0, 1) and every ε > 0, Then, in [12], it was obtained that the function ζ P (s) is analytic in the half-plane σ > α, except for a simple pole at the point s = 1.It turns out that estimates of type (4) are useful for the characterization of the system P.It is known [12] that (1) does not imply the estimate with β 1 < 1.Therefore, together with (1), we suppose that estimate (5) is valid.
Let K be the class of compact subsets of strip D with the connected complement, and H 0 (K) with K ∈ K the class of continuous functions on K that are analytic in the interior of K.Moreover, let L(P ) = {log p : p ∈ P }.
Note, that the following theorem supports the Linnik-Ibragimov conjecture.
Theorem 2. Suppose that the system P satisfies the axioms (1) and (5), and L(P ) is linearly independent over the field of rational numbers Q.Let K ∈ K and f (s) ∈ H 0 (K).Then, for every ε > 0, lim inf Moreover, the limit exists and is positive for all but at most countably many at ε > 0.
Notice that the requirement on the set L(P ) is sufficiently strong, it shows that the numbers of the system P must be different.The simplest example is the system where α is a transcendental number.
An example of P with a bounded mean square is given in [25].
For the proof of Theorem 2, we will build the probabilistic theory of the function ζ P (s) in the space of analytic functions H(D).
The paper is organized as follows.In Section 2, we introduce a certain probability space, and define the H(D) valued random element.Section 3 is devoted to the ergodicity of one group of transformations.In Section 4, we approximate the mean of the function ζ P (s) by an absolutely convergent Dirichlet series.Section 5 is the most important.In this section, we prove a probabilistic limit theorem for the function ζ P (s) on a weakly convergent probability measure in the space H(D), and identify the limit measure.Section 6 gives the explicit form for the support of the limit measure of Section 5.In Section 7, the universality of the function ζ P (s) is proved.

Random Element
Define the Cartesian product The set Ω P consists of all functions ω : P → {s ∈ C : |s| = 1}.In Ω P , the operation of pointwise multiplication and product topology can be defined, and this makes Ω P a topological group.Since the unit circle is a compact set, the group Ω P is compact.Denote by B(X), the Borel σ-field of the space X.Then, the compactness of Ω P implies the existence of the probability Haar measure m P on (Ω P , B(Ω P )), and we have the probability space (Ω P , B(Ω P ), m P ).
Denote the elements of Ω P by ω = (ω(p) : p ∈ P ).Since the Haar measure m P is the product of Haar measures on unit circles, {ω(p) : p ∈ P } is a sequence of independent complex-valued random variables uniformly distributed on the unit circle.
Extend the functions ω(p), p ∈ P, to the generalized integers N P .Let Then we put Now, for s ∈ D and ω ∈ Ω P , define Lemma 1.Under the hypotheses of Theorem 2, ζ P (s, ω) is an H(D)-valued random element defined on the probability space (Ω P , B(Ω P ), m P ).
Proof.Fix σ 0 > σ, and consider Then {a m : m ∈ N P } is a sequence of complex-valued random variables on (Ω P , B(Ω P ), m P ).Denote by z the complex conjugate of z ∈ C. Suppose that m 1 ̸ = m 2 , m 1 , m 2 ∈ N P .Since the set L(P ) is linearly independent over Q, in the product ω(m 1 )ω(m 2 ), there exists at least one factor ω α (p), p ∈ P, with integer α ̸ = 0. Therefore, denoting by Eξ the expectation of the random variable ξ, we have because the integral includes the factor where γ is the unit circle on C, and m γ the Haar measure on γ.This and (7) show that {a m } is a sequence of pairwise orthogonal complex-valued random variables and the series is convergent.Hence, by the classical Rademacher theorem, see [26], the series converges for almost all ω with respect to the measure m P .Therefore, by a property of the Dirichlet series, see [22], the series converges uniformly on compact sets of the half-plane σ > σ 0 for almost all ω ∈ Ω P .Now, let and Denote by the set Ω k ⊂ Ω P such that the series (8) converges uniformly on compact sets of D k for almost all ω ∈ Ω k .Then, by the above remark, On the other hand, taking we obtain from ( 9) that m P ( Ω) = 1, and the series (8) converges uniformly on compact sets of the half-plane σ > σ of the strip D. Hence, ζ P (s, ω) is the H(D)-valued random element on (Ω P , B(Ω P ), m P ).

Lemma 2.
For almost all ω, the product converges uniformly on compact subsets of the half-plane σ > σ, and the equality Proof.The series ζ P (s, ω) is absolutely convergent for σ > 1.Therefore, the equality of the lemma, in view of ( 6), is valid for σ > 1.By proof of Lemma 1, the function ζ P (s, ω), for almost all ω ∈ Ω P , is analytic in the half-plane σ > σ.Therefore, by analytic continuation, it suffices to show that the product of the lemma, for almost all ω ∈ Ω P , converges uniformly on compact subsets of the strip D. Write with We observe that the convergence of product (10) follows from that of the series ∑ p∈P a p (s, ω) and Then Hence, the series is convergent for all ω ∈ Ω P with every σ = σ 0 , σ 0 > σ, thus, uniformly convergent on compact subsets of the half-plane σ > σ.To prove the convergence for the series we apply the same arguments as in the proof of Lemma 1.For fixed σ > σ, we have and for p, q ∈ P, converges for almost all ω ∈ Ω P .Hence, this series, for almost all ω ∈ Ω P , converges uniformly on compact subsets of the half-plane σ > σ.This, together with a convergence property of the series (11), shows that the series for almost all ω ∈ Ω P , converges uniformly on compact subsets of the half-plane σ > σ, and it remains to prove the same for the series Clearly, for all ω ∈ Ω P , Hence, the series (12), for all ω ∈ Ω P , converges uniformly on compact subsets of the half-plane σ > σ.

Ergodicity
For τ ∈ R, let κ τ = p −iτ : p ∈ P , and Since the Haar measure m P is invariant with respect to shifts by elements of Ω P , i.e., for all A ∈ B(Ω P ) and ω ∈ Ω P , m P (A) = m P (ωA) = m P (Aω), g τ (m) is a measurable measure preserving transformation on Ω P .Thus, we have the one-parameter group G τ = {g τ : τ ∈ R} of transformations of Ω P .A set A ∈ B(Ω P ) is called invariant with respect to G τ if, for every τ ∈ R, the sets A and A τ = g τ (A) differ one from another at most by a set of m P -measure zero.It is well known that all invariant sets form a σ-field which is a subfield of B(Ω P ).The group G τ is called ergodic if its σ-field of invariant sets consists only of sets m P -measure 0 or 1. Lemma 3.Under the hypotheses of Theorem 2, the group G τ is ergodic.
Proof.Let A ∈ B(Ω P ) be a fixed invariant set of G τ .Denote by I A (ω) the indicator function of the set A. Then, for almost all ω ∈ Ω P , Characters χ of the group Ω P are of the form where * indicates that only a finite number of integers k p are distinct from zero.Suppose that χ is a nontrivial character, i.e., χ(ω) ̸ ≡ 1 for all ω ∈ Ω P .Then, we have Since the set L(P ) is linearly independent over Q, and χ is a nontrivial character, Thus, there exists a real number a ̸ = 0 such that Hence, there is τ 0 ∈ R satisfying χ(g τ 0 ) ̸ = 1.Now, we deal with Fourier analysis on Ω P .Denote by g the Fourier transform of a function g, i.e., In virtue of (13), we find Hence, in view of inequality χ(g τ 0 ) ̸ = 1, we obtain Consider the case of the trivial character χ 0 of the group Ω P .We set I A (χ 0 ) = c.Then, the orthogonality of characters implies that Therefore, using (15) yields the equality It is well known that a function is completely determined by its Fourier transform.Thus, by (16), we have that for almost all ω ∈ Ω P , I A (ω) = c.However, as I A (ω) is the indicator function, it follows that c = 0 or 1.In other words, for almost all ω ∈ Ω P , I A (ω) = 0 or I A (ω) = 1.Thus, m P (A) = 0 or m P (A) = 1.The lemma is proved.
We apply Lemma 3 for the estimation of the mean square for ζ P (s, ω).Proof.Let a m (σ, ω), m ∈ N P , be the same as the proof of Lemma 1.The random variables a m (σ, ω) are pairwise orthogonal, and Therefore, Let g τ (ω) be the transformation from the proof of Lemma 3.Then, by the definition of g τ , We recall that a strongly stationary random process X(t, ω), t ∈ T , on (Ω, A, P) is called ergodic if its σ-field of invariant sets consists of sets of P-measure 0 or 1.Since the group G τ is ergodic, the stationary process |ζ P (σ + it, ω)| 2 is ergodic, for details, see [22].Therefore, we can apply the classical Birkhoff-Khintchine ergodic theorem, see [27].This gives, by (17), lim

Approximation in the Mean
In this section, we approximate the functions ζ P (s) and ζ P (s, ω) by absolutely convergent Dirichlet series.Let η > 1 − σ be a fixed number, and, for m ∈ N P and n ∈ N, Then the series , ω ∈ Ω P , are absolutely convergent for σ > σ and for every fixed n ∈ N.
and every compact set K ⊂ D lies in some K l .Then is the desired metric in H(D).
In [24], the following statement has been obtained.
Lemma 5. Suppose that (1) is valid.Then Denote by Ω P,1 a subset of Ω P such that a product converges uniformly on compact subsets of D for ω ∈ Ω P,1 , and by Ω P,2 a subset of Ω P such that, for ω ∈ Ω P,2 , the estimate Then, by Lemmas 3 and 4, m P (Ω P,j ) = 1, j = 1, 2. Let Then again m P ( Ω P ) = 1.

Limit Theorems
In previous sections, we gave preparatory results for the proof of a limit theorem for ζ P (s) in the space of analytic functions H(D).In this section, we consider the weak convergence for We start with a limit lemma on Ω P .For A ∈ B(Ω P ), define Lemma 7. Suppose that the set L(P ) is linearly independent over Q.Then P Ω P T,P converges weakly to the Haar measure m P as T → ∞.
Proof.In the proof of Lemma 3, we have seen that characters of the group Ω P are given by (14).Therefore, the Fourier transform F T,P (k), k = (k p : k p ∈ Z, p ∈ P ) of P Ω P T,P is defined by We have to show that lim For this, we apply the linear independence of the set L(P ).We have if and only if k p = 0. Thus, (24), otherwise, and ( 25) take place.
The next lemma is devoted to the functions ζ P,n (s) and ζ P,n (s, ω).For A ∈ B(H(D)), set Lemma 8. Suppose that the set L(P ) is linearly independent over Q.Then, on (H(D), B(H(D))) there exists a probability measure P P,n such that both the measures P T,P,n and P T,P,n converge weakly to P P,n as T → ∞.
Proof.We use a property of the preservation of weak convergence under continuous mappings.Consider the mapping v P,n : Ω P → H(D) given by v P,n (ω) = ζ P,n (s, ω).
Since the series for ζ P,n (s, ω) is absolutely convergent for σ > σ, the mapping v P,n is continuous.Moreover, for A ∈ B(H(D)), Thus, denoting by P Ω P T,P v −1 P,n the measure given by the latter equality, we obtain that P T,P,n = P Ω P T,P v −1 P,n .This equality continuity of v P,n , and the principle of preservation of weak convergence, see Theorem 5.1 of [28], show that P T,P,n converges weakly to the Then, repeating the above arguments, we find that P T,P,n converges weakly to Then, by invariance of the measure m P , we have Thus, P T,P,n and P T,P,n converge weakly to the same measure Q P,n as T → ∞.
Next, we study the family of probability measures {Q P,n : n ∈ N}.We recall some notions.A family of probability measures {P} on (X, B(X)) is called tight if, for every ε > 0, there exists a compact set K ⊂ X such that P(K) > 1 − ε for all P, and {P} is relatively compact if every sequence {P k } ⊂ {P} has a subsequence {P n k } weakly convergent to a certain probability measure P on (X, B(X)) as k → ∞.By the classical Prokhorov theorem, see Theorem 6.1 of [28], every tight family of probability measures is relatively compact.Lemma 9.Under the hypotheses of Theorem 2, the family {Q P,n : n ∈ N} is relatively compact.
Proof.In view of the above remark, it suffices to prove the tightness of {Q P,n }.Let K ⊂ D be a compact.Then, using the Cauchy integral formula and absolute convergence of the series for ζ P,n (s), we obtain σ κ > σ Suppose that ξ T is a random variable on a certain probability space (Ξ, A, µ) uniformly distributed in the interval [0, T].Define the H(D)-valued random element Now, let K = K l , where {K l } is a sequence of compact sets of D from the definition of the metric ρ.Fix ε > 0, and set R l = 2 l ε −1 √ V l where V l = V κ l .Therefore, relation (26), and the Chebyshev type inequality yield lim sup Hence, in view of (28), Define the set Then H(ε) is a compact set in H(D).Moreover, inequality (29) implies that is the distribution of Y P,n , this shows that The lemma is proved.Now, we are ready to consider the weak convergence for P T,P and P T,P .For convenience, we recall one general statement.Proposition 1. Suppose that a metric space (X, d) is separable, and the X-valued random elements x mn and y n , m, n ∈ N are defined on the same probability space (Ξ, A, µ).Suppose that Proof.The proposition is Theorem 4.2 of [28], where its proof is given.
Lemma 10.Under the hypotheses of Theorem 2, on (H(D), B(H(D))) there exists a probability measure P P such that both the measures P T,P and P T,P converge weakly to P P as T → ∞.Proof.We will show that the limit measure Q P in Lemma 10 coincides with P ζ P .

Proof
We apply the equivalent of weak convergence of probability measures in terms of continuity sets, see Theorem 2.1 of [28].Let A be a continuity set of the measure Q P , i.e., Q P (∂A) = 0, where ∂A denotes the boundary of A. Then, Lemma 10 implies that lim T→∞ P T,P (A) = Q P (A). (34) On (Ω P , B(Ω P )), define the random variable Return to the group G τ of Lemma 3. Since, by Lemma 3, the group G τ is ergodic, the process ξ(g τ (ω)) is ergodic, and application of the Birkhoff-Khintchine theorem [27] gives for almost all ω ∈ Ω P .However, the definition of the random variable ξ T (ω) implies that, for almost all ω ∈ Ω P , 1 Thus, by (34), lim Moreover, This, (35) and (36) prove that Q P (A) = P ζ P (A) for all continuity sets A of the measure Q P .It is well known that all continuity sets constitute a determining class.Hence, we have Q P = P ζ P , and the theorem is proved.

Support
For the proof of Theorem 2, the explicitly given support of the measure P ζ P is needed.We recall that the support of P ζ P is a minimal closed set S P ⊂ H(D) such that P ζ P (S P ) = 1.Every open neighbourhood of elements S P has a positive P ζ P -measure.
Proposition 2. Under the hypotheses of Theorem 2, the support of the measure P ζ P is the set S P .
A proof of Proposition 2 is similar to that in the case of the Riemann zeta-function.Therefore, we will state without proof only the lemmas because their proofs word for word coincide with analogical assertions from [22].
We start with some estimations over generalized primes p ∈ P.
Proof.We have Recall that the support of the distribution of a random element X is called a support of X, and is denoted S X .
For convenience, we state a lemma on the support of a series of random elements.
Lemma 14.Let {ξ m } be a sequence of independent H(D)-valued random elements on a certain probability space (Ξ, A, µ); the series ∞ ∑ m=1 ξ m is convergent almost surely.Then, the support of the sum of this series is the closure of the set of all g ∈ H(D) which may be written as a convergent series Proof.The lemma is Theorem 1.7.10 of [22], where its proof is given.
On the other hand, the random element ζ P (s, ω) is convergent for almost all ω ∈ Ω P , a product of non-zeros multipliers.Therefore, by the classical Hurwitz theorem, see [29], This inclusion together with (37) proves the proposition.

Proof of Universality
In this section, we prove Theorem 2. Its proof is based on Theorem 3, Proposition 2 and the Mergelyan theorem [30] on the approximation of analytic functions by polynomials on compact sets with connected complements.
Proof of Theorem 2. Let p(s) be a polynomial, K and ε defined in Theorem 2, and G ε = g ∈ H(D) : sup s∈K g(s) − e p(s) < ε 2 .
Then, the set G ε is an open neighborhood of an element e p(s) ∈ S P .Since, view of Proposition 2, S P is the support of the measure P ζ P , by a property of supports, we have Hence, the boundaries ∂ G ε 1 and ∂ G ε 2 do not intersect for different positive ε 1 and ε 2 .Therefore, P ζ P (∂ G ε ) > 0 for countably many ε > 0. In other words, the set G ε is a continuity set of the measure P ζ P for all but at most countably many ε > 0. This, (39), Theorem 3 and the equivalent of weak convergence in terms of continuity sets prove the second statement of the theorem.

Conclusions
In the paper, we considered the set P of generalized prime numbers satisfying ∑ m⩽x m∈N P 1 = ax + O x β , a > 0, 0 ⩽ β < 1, and where N P is the set of generalized integers and Λ P (m) is the generalized von Mangoldt function corresponding to the set P. Assuming that the set {log p : p ∈ P } is linearly independent over Q, and the Beurling zeta-function has the bounded mean square for σ > σ with some β < σ < 1, we obtained universality of ζ P (s), i.e., that every non-vanishing analytic function can be approximated by shifts ζ P (s + iτ), τ ∈ R.
Thus, the series ∑ p∈P E|b p (σ, ω)| 2 log 2 p is convergent, and the Rademacher theorem implies that the series ∑ p∈P b p (σ, ω)

) 2 .
Since f (s) ∈ H 0 (K), we may apply the mentioned Mergelyan theorem and choose the polynomial p(s) satisfying sup s∈K f (s) − e p(s) < ε This shows that the set G ε lies inG ε = g ∈ H(D) : sup s∈K |g(s) − f (s)| < ε .Thus, by (38), we haveP ζ P ( G ε ) > 0. (39)Theorem 3 and the equivalent of weak convergence in terms of open sets yield lim infT→∞ P T,P ( G ε ) ⩾ P ζ P ( G ε ).This, (39), and the definitions of P T,P and G ε prove the first statement of the theorem.To prove the second statement of the theorem, we observe that the boundary ∂ G ε of the set G ε lies in the set g ∈ H(D) : sup s∈K |g(s) − f (s)| = ε .
|a p | = 1.By Lemma 13, the set of the latter series is dense in H(D).Define u : H(D) → H(D) by u(g) = e g , g ∈ H(D).The mapping u is continuous, u(log ζ P (s, ω)) = ζ P (s, ω) and u(H(D)) = S P \ {0}.This shows that S P \ {0} lies in the support of ζ P (s, ω).Since the support is a closed set, we obtain that the support of ζ P (s, ω) contains the closure of S P \ {0}, i.e., S ζ P ⊃ S P .( Proof of Proposition 2. By the definition, {ω(p) : p ∈ P } is a sequence of independent complex-valued random variables.Therefore, {g P (s, ω(p))} is a sequence of independent H(D)-valued random elements.Since the support of each ω(p) is the unit circle, the support of g P (s, ω(p))} is the setg ∈ H(D) : g(s) = − log 1 − a p s , |a| = 1 .Therefore, in view of Lemma 14, the support of the H(D)-valued random element log ζ P (s, ω) = − ∑ with