Joint universality of the Riemann zeta-function and Lerch zeta-functions

. In the paper, we prove a joint universality theorem for the Riemann zeta-function and a collection of Lerch zeta-functions with parameters algebraically independent over the ﬁeld of rational numbers.


Introduction
Let λ ∈ R and α, 0 < α 1, be fixed parameters.The Lerch zeta-function L(λ, α, s), s = σ + it, is defined, for σ > 1, by For λ ∈ Z, the function L(λ, α, s) reduces to the Hurwitz zeta-function ζ(s, α) which is a meromorphic function with a unique simple pole at the point s = 1 with residue 1.If λ / ∈ Z, then the Lerch zeta-function has analytic continuation to an entire function.In view of the periodicity of e 2πiλm , we can suppose that 0 < λ 1.
It is well known that the Lerch zeta-function L(λ, α, s) with transcendental parameter α is universal (see [1], also [2]).Let D = {s ∈ C: 1/2 < σ < 1}.Denote by K the class of compact subsets of the strip D with connected complements, and, for K ∈ K, denote by H(K) the set of continuous functions on K which are analytic in the interior of K.
Moreover, we use the notation meas{A} for the Lebesgue measure of a measurable set A ⊂ R. Then the universality of L(λ, α, s) is contained in the following theorem.
Theorem 1. Suppose that α is transcendental.Let K ∈ K and f (s) ∈ H(K).Then, for every > 0, lim inf Thus, the universality of L(λ, α, s) means that the shifts L(λ, α, s + iτ ) approximate with a given accuracy a wide class of analytic functions.
In [9], a joint universality theorem for the Riemann zeta-function ζ(s) and periodic Hurwitz zeta-functions has been obtained.Let A = {a m : m ∈ N 0 } be a periodic sequence of complex numbers with minimal period k ∈ N. We remind that the periodic Hurwitz zeta-function ζ(s, α; A) with parameter α, 0 < α 1, is defined, for σ > 1, by the Dirichlet series and is meromorphically continued to the whole complex plane with a unique possible pole at the point s = 1 with residue If a = 0, then ζ(s, α; A) is an entire function.has been proved.Here a collection of periodic sequences A jl , A jl = {a mjl : m ∈ N 0 }, with minimal period k jl ∈ N, l = 1, . . ., l j , corresponds the parameter α j , 0 < α j 1, j = 1, . . ., r.For K ∈ K, denote by H 0 (K) the class of continuous non-vanishing functions on K which are analytic in the interior of K. Let k j be the least common multiple of the periods k j1 , . . ., k jlj , and Then the main result of [9] is of the form.
Theorem 3. Suppose that the numbers α 1 , . . ., α r are algebraically independent over Q, and that rank(A j ) = l j , j = 1, . . ., r.For j = 1, . . ., r and l = 1, . . ., l j , let K jl ∈ K and f jl ∈ H(K jl ).Moreover, let K ∈ K and f (s) ∈ H 0 (K).Then, for every > 0, lim inf We call the approximation property of the functions (1) in Theorem 3 a mixed joint universality because the function ζ(s) and the functions ζ(s, α j ; A jl ) are of different types: the function ζ(s) has Euler product, while the functions ζ(s, α j ; A jl ) with transcendental α j do not have Euler product over primes.This is reflected in the approximated functions: the function f (s) must be non-vanishing on K, while the functions f jl are arbitrary continuous functions on K jl .
The first mixed joint universality theorem has been obtained by Mishou [10] for the Riemann zeta-function and Hurwitz zeta-function ζ(s, α) with transcendental parameter α.This result in [11] has been generalized for a periodic zeta-function and a periodic Hurwitz zeta-function.In [12], the latter mixed joint universality theorem has been extended for several periodic zeta-functions and periodic Hurwitz zeta-functions.
Universality theorems for zeta-functions have a series of interesting applications.From them, for example, various denseness results of Bohr's type for values of zetafunctions follow.The universality implies the functional independence of zeta-functions.This property of zeta-functions is applied to the zero-distribution of those zeta-functions.In [13], the universality has been applied to the famous class number problem.Universality theorems find applications even in solving some problems of physics [14].For the above mentioned and other facts related to universality and references, we refer to [2,[15][16][17][18][19][20].
Thus, the universality of zeta-functions is a very interesting and useful property which motivates to continue investigations in the field.
The aim of this paper is to replace the zeta-functions ζ(s, α j ; A jl ) with periodic coefficients in Theorem 3 by Lerch zeta-functions L(λ j , α j , s) with arbitrary λ j ∈ (0, 1] whose coefficients, in general, are not periodic.This is the novelty of the paper. Theorem 4. Suppose that the numbers α 1 , . . ., α r are algebraically independent over Q.For j = 1, . . ., r, let λ j ∈ (0, 1], K j ∈ K and f j ∈ H(K j ).Moreover, let K ∈ K and f (s) ∈ H 0 (K).Then, for every > 0, lim inf We note that the linear independence of the set L(α 1 , . . ., α r ) is not sufficient for the proof of Theorem 4 because we need the linear independence of the set where P is the set of all prime numbers.This set consists of logarithms of all prime numbers and of all logarithms log(m + α j ), m ∈ N, j = 1, . . ., r. Really, L is a multiset.For example, if L has two identical elements, then it is linearly dependent over Q.The proof of Theorem 4 is based on a joint limit theorem on weakly convergent probability measures in the space of analytic functions.
Let H(D) be the space of analytic functions on D endowed with the topology of uniform convergence on compacta, and let r 1 = r + 1.On the probability space (Ω, B(Ω), m H ), define the H r1 (D)-valued random element ζ(s, α, λ, ω) by the formula and e 2πiλj m ω j (m) (m + α j ) s , j = 1, . . ., r.
Now we state a limit theorem on the space (H r1 (D), B(H r1 (D))).
Then Q T converges weakly to the Haar measure m H as T → ∞.
Proof.The proof of the lemma is given in [9, Lemma 1].
Proof.The proof uses Lemma 1 and does not depend on the coefficients of the functions L n (λ j , α j , s), j = 1, . . ., r.Therefore, it coincides with the proof of [9, Lemma 2].
Now we define a metric on H r1 (D) which induces the topology of uniform convergence on compacta.For g 1 , g 2 ∈ H(D), we define where {K m : m ∈ N} is a sequence of compact subsets of the strip D such that The existence of the sequence {K m } follows from a general theorem, see, for example, [21], however, in the case of the region D, it is easily seen that we can take closed rectangles.Clearly, ρ is a metric on H(D) inducing its topology.For g j = (g j , g j1 , . . ., g jr ) ∈ H r1 (D), j = 1, 2, we put Then we have that ρ is a desired metric on H r1 (D).Using this metric, we approximate Moreover, suppose that the numbers α 1 , . . ., α r are algebraically independent over Q.
Then, for almost all ω ∈ Ω, Proof.In [16], it is proved that and, for almost all ω ∈ Ω lim n→∞ lim sup Since the numbers α 1 , . . ., α r are algebraically independent over Q, each number α j is transcendental.Therefore, in [2], it was obtained that, for j = 1, . . ., r, and, for almost all ω j ∈ Ω j , All these equalities together with the definition of the metric ρ prove the lemma.
Then P T and PT both converge weakly for almost all ω ∈ Ω to the same probability measure P on (H r1 (D), B(H r1 (D))) as T → ∞.
Proof.We give a shortened proof because we apply similar arguments as in [9].Let θ be a random variable defined on a certain probability space (Ω 0 , A, P) and uniformly distributed on Then, in view of Lemma 2, X T,n X n , where X n is the random element with the distribution P n (P n is the limit measure in Lemma 2), and D → denotes the convergence in distribution.Using the absolute convergence of series for ζ n (s) and L n (λ j , α j , s), j = 1, . . ., r, we prove without difficulties that the family of probability measures {P n : n ∈ N} is tight.Hence, by the Prokhorov theorem, this family is relatively compact.Thus, we have a subsequence {P n k } such that P n k converges weakly to some probability measure P as k → ∞.Hence, Then Lemma 3 implies that, for every > 0, lim n→∞ lim sup This, (2), (3) and Theorem 4.2 of [22] show that Then {Φ τ : τ ∈ R} is an ergodic group of measurable measure preserving transformations on Ω (see [12]).
Let ξ be a random variable on (Ω, B(Ω), m H ) given by where A is a fixed continuity set of the measure P .By Lemma 4, for almost all ω ∈ Ω, The ergodicity of the group {Φ τ : τ ∈ R} implies that of the process ξ(Φ τ (ω)).Therefore, the classical Birkhoff-Khintchine theorem shows that, for almost all ω ∈ Ω, where Eξ denotes the expectation of ξ.The definitions of ξ and of Φ τ give the equalities In view of (7), Hence, obviously, P ζ (S) = 1.Moreover, if A ∈ B(H(D)) with A S, or A j ∈ B(H(D)) with A j H(D), for some j, then, in view of the minimality of S and H(D) for P ζ (A) and P Lj (A j ), respectively, we have that P ζ (A) < 1 or P Lj (A j ) < 1.Thus, then P ζ (B) < 1.Hence, the minimality of S follows.

Universality theorem
In this section, we will prove Theorem 4. Its proof is based on Theorems 5 and 6 as well as on the Mergelyan theorem on the approximation of analytic functions by polynomials.We state this theorem as the next lemma.Lemma 5. Let K ⊂ C be a compact set with connected complement, and f (s) be a continuous function on K which is analytic in the interior of K.Then, for every > 0, there exists a polynomial p(s) such that Proof.The proof of the lemma can be found in [23], see also [24].
Proof of Theorem 4. By Lemma 5, there exists a polynomial p(s) such that Since f (s) = 0 on K, p(s) = 0 on K as well provided is small enough.Thus, we can define on K a continuous branch of log p(s) which will be analytic in the interior of K.
Applying Lemma 5 once more, we obtain that there exists a polynomial q(s) such that sup s∈K p(s) − e q(s) < 4 .
This together with (8) shows that sup s∈K f (s) − e q(s) < 2 .
Again, by Lemma 5, there exist polynomials p j (s) such that Combining this with (11) gives the assertion of the theorem.