Approximation by Shifts of Compositions of Dirichlet L -Functions with the Gram Function

: In this paper, a joint approximation of analytic functions by shifts of Dirichlet L -functions L ( s + ia 1 t τ , χ 1 ) , . . . , L ( s + ia r t τ , χ r ) , where a 1 , . . . , a r are non-zero real algebraic numbers linearly independent over the ﬁeld Q and t τ is the Gram function, is considered. It is proved that the set of their shifts has a positive lower density.


Introduction
Let χ : N → C be a Dirichlet character modulo q ∈ N. Note that χ(m) is periodic with period q, completely multiplicative (i.e., χ(mn) = χ(m)χ(n) for all m, n ∈ N and χ(1) = 1), χ(m) = 0 for (m, q) = 1 and χ(m) = 0 for (m, q) = 1. Let s = σ + it. In [1], L. Dirichlet introduced a function which is now called the Dirichlet L-function. In virtue of the complete multiplicativity of χ(m), the function (1) can be written as an Euler product where P is the set of all prime numbers and has a meromorphic continuation to the whole complex plane with a unique simple pole at the point s = 1 (if χ is the principal character modulo q) with residue ∏ p|q (1 − 1/p). Since then, the function (1) has become a subject of intensive investigation. See, for instance, References [2][3][4] for some very recent papers on its zeros and moments. For q = 1, the function L(s, χ) becomes the Riemann zeta-function ζ(s).
The Voronin theorem was extended to more general compact sets independently in References [6][7][8].
Denote by K the class of compact subsets of the strip D = {s ∈ C : 1/2 < σ < 1} with connected complements, and by H 0 (K), where K ∈ K, the class of continuous non-vanishing functions on K that are analytic in the interior of K. Then the modern version of the Voronin theorem asserts that if K ∈ K and f (s) ∈ H 0 (K), then, for every ε > 0, lim inf where measA stands for the Lebesgue measure of a measurable set A ⊂ R (see, for example, Reference [9]). The latter inequality shows that there are infinitely many shifts L(s + iτ, χ) approximating a given function from the class H 0 (K).
In Reference [10], Voronin considered the joint functional independence of Dirichlet L-functions using the joint universality. We recall that two Dirichlet characters are called non-equivalent if they are not generated by the same primitive character. Thus, the following statement is valid [10,11]; see also References [9,12,13]. Theorem 1. Let χ 1 , . . . , χ r be pairwise non-equivalent Dirichlet characters. For j = 1, . . . , r, let K j ∈ K, and f j (s) ∈ H 0 (K j ). Then, for every ε > 0, The non-equivalence of the characters χ 1 , . . . , χ r ensures a certain independence of the functions L(s, χ 1 ), . . . , L(s, χ r ) which is necessary for a simultaneous approximation of the collection f 1 (s), . . . , f r (s). Later, it turned out that, in place of non-equivalent characters, different shifts can be used. This was observed by Nakamura [14]. More precisely, he proved the following theorem. Theorem 2. Let a 1 = 1, a 2 , . . . , a r be real algebraic numbers linearly independent over the field of rational numbers Q and χ 1 , . . . , χ r be arbitrary Dirichlet characters. For j = 1, . . . , r, let K j ∈ K, and let f j (s) ∈ H 0 (K j ). Then, for every ε > 0 and a ∈ R \ {0}, In Reference [15], Pańkowski obtained the joint universality of Dirichlet L-functions using the shifts L(s + iα j τ a j log b j τ, χ j ), j = 1, . . . , r, where α 1 , . . . , α r ∈ R, a 1 , . . . , a r ∈ R + are distinct, b 1 , . . . , b r are distinct and satisfy The aim of this paper is to introduce new shifts of Dirichlet L-functions that approximate collections of analytic functions from the class H 0 (K). Let, as usual, Γ(s) be the Euler gamma-function. For t > 0, denote the increment θ(t) of the argument of the function π −s/2 Γ (s/2) along the segment connecting the points s = 1/2 and s = 1/2 + it. Then it is known (see, for example, Reference [16] [Lemma 1.1]) that, for τ ≥ 0, the equation has the unique solution t τ satisfying θ (t τ ) > 0. For n ∈ N, the numbers t n are called the Gram points. They were introduced and studied in Reference [17]. Therefore, we call t τ the Gram function. A very interesting property of the Gram points is the relation t n ∼ γ n as n → ∞, where γ n > 0 are imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we will consider the joint approximation of analytic functions by shifts of Dirichlet L-functions involving the Gram function. More precisely, we will prove the following joint universality theorem. Theorem 3. Suppose that a 1 , . . . , a r are real non-zero algebraic numbers linearly independent over Q, and χ 1 , . . . , χ r are arbitrary Dirichlet characters. For j = 1, . . . , r, let K j ∈ K and f j (s) ∈ H 0 (K j ). Then, for every Moreover, the limit exists for all but at most countably many ε > 0.
For the proof of Theorem 3, we will use the probabilistic approach based on weakly convergent probability measures in the space of analytic functions.

Lemmas
We start with a lemma on the functional properties of the function t τ . (Its proof can be found in Reference [16] [Lemma 1.1].) and t τ = − π τ(log τ) 2 The next lemma provides an estimate for certain trigonometric integral.

Lemma 2.
Suppose that F(x) is a real differentiable function, the derivative F (x) is monotonic and The proof of the lemma is given, for example, in Reference [11]. We will also use Baker's theorem on linear forms in logarithms of algebraic numbers (see, for example, Reference [18]).
Then Ω r is also a compact topological Abelian group. Therefore, denoting by B(X) the Borel σ-field of the space X, we see that, on (Ω r , B(Ω r )), the probability Haar measure m r H exists. This gives the probability space (Ω r , B(Ω r ), m r H ).
Then the following limit theorem holds.

Lemma 4.
Under hypotheses of Theorem 2 on the numbers a 1 , . . . , a r , Q T converges weakly to the Haar measure m r H as T → ∞.
Proof. We apply the Fourier transform method. It is well known that the dual group of Ω r is isomorphic to the group where Z jp = Z for all j = 1, . . . , r, p ∈ P. Hence it follows that characters of the group Ω r are of the form where ω j (p) is the pth component of an element ω j ∈ Ω j , j = 1, . . . , r, and the sign " * " means that only a finite number of integers k jp are distinct from zero. Therefore is the Fourier transform of a measure µ on (Ω r , B(Ω r )).
. . , r, be the Fourier transform of Q T . In view of (2) we have Thus, by the definition of Q T , Obviously, if k = (0, . . . , 0), then Now suppose that k = (k 1 , . . . , k r ) = (0, . . . , 0). Note that Since k j = 0 for some j ∈ {1, 2, . . . , r}, there is a prime number p such that k jp = 0. For this p, the sum β p de f = ∑ r j=1 a j k jp is non-zero, because the numbers a 1 , . . . , a r are linearly independent over Q. It is well known that the set {log p : p ∈ P} is linearly independent over Q. Therefore, in view of Lemma 3, Now, (3) and Lemmas 1 and 2 show that, in the case k = (0, . . . , 0), This together with (4) and (5) give Since the right-hand side of the latter equality is the Fourier transform of the Haar measure m r H , the lemma follows by a continuity theorem for probability measures on compact groups.
Moreover, let u n : Ω r → H r (D) be given by the formula The absolute convergence of the series for L n (s, ω j , χ j ) implies the continuity of the mapping u n .
In view of (7) and (8) we conclude that Lemma 4, the continuity of u n and the well-known property on preservation of weak convergence under mapping lead to the following statement.

Lemma 5.
Under hypothesis of Theorem 3 on the numbers a = (a 1 , . . . , a r ), we have converges weakly to the measure V n as T → ∞.
The probability measure V n is very important for the proof of Theorem 3. Let Note that the latter products are uniformly convergent on compact subsets of the strip D for almost all ω j ∈ Ω j , and define the H(D)-valued random elements on the probability space (Ω j , B(Ω j ), m jH ), where m jH is the probability Haar measure on (Ω j , B(Ω j )). Therefore, L(s, ω, χ) is the H r (D)-valued random element on (Ω r , B(Ω r ), m r H ). Denote by P L the distribution of the random element L(s, ω, χ), that is, P L (A) = m r H ω ∈ Ω r : L(s, ω, χ) ∈ A , A ∈ B(H r (D)).
We recall that the support of a probability measure P on (X, B(X)), where the space X is separable, is a minimal closed set S P ⊂ X such that P(S P ) = 1. The set S P consists of all elements x ∈ X such that, for every open neighbourhood G of x, the inequality P(G) > 0 is satisfied.
The measure V n is independent on any hypothesis. Therefore, from Reference [19] it follows that: Lemma 6. The measure V n converges weakly to P L as n → ∞. Moreover, the support of P L is the set S r , where Proof. To be precise, in Reference [19] it was proved that a certain measure P N converges weakly to a certain probability measure P on (H r (D), B(H r (D))) (as N → ∞), and the measure P is the limit measure of V n as n → ∞. Moreover, it was proved that P = P L . It remains to prove that the support of P L is the set S r . It is well known that the support of the random element is the set S for every Dirichlet character χ. Since the space H r (D) is separable, we have
Since the Haar measure m r H is the product of the Haar measures m jH on (Ω j , B(Ω j )), j = 1, . . . , r, we deduce that This equality and the minimality of the support together with remark on the support of the element (10) show that the support of P L is the set S r .
To pass from L n (s + iat τ , χ) (defined by (7)) to L(s + iat τ , χ), certain mean square estimates for Dirichlet L-functions are necessary. Let χ be an arbitrary character modulo q.
For g 1 , g 2 ∈ H(D), define where {K l } ⊂ D is a sequence of compact subsets such that D = ∞ ∪ l=1 K l , K l ⊂ K l+1 for all l ∈ N, and if K ⊂ D is a compact set, then K ⊂ K l for some l ∈ N. Then ρ is a metric in the space H(D) inducing the topology of uniform convergence on compacta. Now, putting, for g 1 = (g 11 , . . . , g 1r ), g 2 = (g 21 , . . . , g 2r ) ∈ H r (D), gives a metric in H r (D) inducing the product topology. The next lemma provides a certain approximation of L(s, χ) (see definition (11)) by L n (s, χ).

Lemma 8.
Suppose that a = (0, . . . , 0). Then Proof. From the definition (13) of the metric ρ, it follows that it suffices to prove that, for a = 0, for every j = 1, . . . , r. We will prove the above equality for the character χ modulo q.
Let θ be from the definition (6) of v n (m), and Then the representation is true. Its proof is the same as in Section 5.4 of [21] for the Riemann zeta-function. Hence, taking θ 1 > 0, by the residue theorem, we obtain where Let K ⊂ D be an arbitrary compact set. Denote by s = σ + iv the points of K, and suppose that 1/2 + 2ε ≤ σ ≤ 1 − ε with fixed ε > 0 for s ∈ K. More precisely, we select Then, in view of (16), Now, taking t in place of t + v, we get that, for s ∈ K, This implies the estimate It is well known that uniformly in σ, σ 1 ≤ σ ≤ σ 2 , with arbitrary σ 1 < σ 2 , Therefore, by the definition (15) of the function l n (s), we find that, for s ∈ K, In the same way, for s ∈ K, we obtain Suppose that θ = 1/2 + ε. Then (17), (19) and Lemma 7 lead to the bound Moreover, by (18), Lemma 1 and (20), Thus, in view of (17) and (21), From this, it follows that lim n→∞ lim sup Now, the definition (12) of the metric ρ implies (14), which completes the proof of Lemma 8.

A Limit Theorem
For A ∈ B(H r (D)), define In this section, we will prove the following statement.

Theorem 4.
Suppose that a 1 , . . . , a r are non-zero real algebraic numbers linearly independent over Q, and χ 1 , . . . , χ r are arbitrary Dirichlet characters. Then P T converges weakly to P L as T → ∞. The support of P L is the set S r .
First we recall a useful property of convergence in distribution ( D − →) (see Theorem 4.2 in Reference [20]).

Lemma 9.
Suppose that the space (X, d) is separable, the random elements X kn and Y n , k ∈ N, n ∈ N, are defined on the same probability space with measure µ, for every k ∈ N, and, for every ε > 0, lim In the theory of weak convergence of probability measures, the notions of relative compactness and tightness of families of probability measures are very useful. We recall that the family {P} of probability measures on (X, B(X)) is called relatively compact if every sequence {P n } ⊂ {P} contains a weakly convergent subsequence to a certain measure on (X, B(X)), and this family is called tight, if for every ε > 0, there exists a compact set K = K(ε) ⊂ X such that P(K) > 1 − ε for all P ∈ {P}. By the direct Prokhorov theorem (see Theorem 5.1 in Billingsley [20]), every tight family {P} is relatively compact. We apply the above remarks to the sequence {V n : n ∈ N}, where V n (defined by (8)) is the limit measure in Lemma 5.
Lemma 10. The sequence {V n } is relatively compact.
Proof. By the above mentioned Prokhorov theorem, it suffices to prove that the sequence {V n } is tight.
Suppose θ T is a random variable defined on a certain probability space with measure µ and uniformly distributed on [2, T]. Define the H r (D)-valued random element X T,n = X T,n (s) = (X T,n,1 (s), . . . , X T,n,r (s)) = L n (s + iat θ T , χ).

Moreover, let
be the H r (D)-valued random element with the distribution V n . Then Lemma 5 implies the relation By Lemma 7 with t = 0, we have, for 1/2 < σ < 1, Let K l be a compact set from the definition of the metric ρ. Then (26) together with the Cauchy integral formula show that T 2 sup s∈K l |L(s + ia j t τ , χ j )| dτ l,a j T, j = 1, . . . , r.
This combined with (22) implies the inequality sup n∈N lim sup Fix ε > 0, and define M lj = M lj (s) = 2 l rR lj ε −1 . Then, in view of (27), we find that, for each n ∈ N, This together with (25) shows that, for all l, n ∈ N, Define the set Then K j is a compact set in H(D), and, in virtue of (24) and (28), In other words, we have V n (K) ≥ 1 − ε for all n ∈ N. Thus, the sequence {V n : n ∈ N} is tight.
Proof of Theorem 4. By Lemma 10, there exists a subsequence {V n k } of the sequence {V n } that is weakly convergent to a certain probability measure P on (H r (D), B(H r (D))) as k → ∞. This can be written as Define one more H r (D)-valued random element Then Lemma 8 implies that, for every ε > 0, The latter equality together with (25), (29), and Lemma 9 shows that or, in other words, P T converges weakly to P as T → ∞. Moreover, by the relation (30), the measure P is independent of the subsequence {V n k }. Thus, we deduce that or V n converges weakly to P as n → ∞. Therefore, the theorem follows by Lemma 6.

Proof of Universality
The proof of Theorem 3 is based on Mergelyan's theorem on the approximation of analytic functions by polynomials [22], Theorem 4, and the properties of weak convergence. For convenience, we state them as lemmas.
Lemma 11 (Mergelyan theorem). Suppose that K ⊂ C is a compact set with connected complement, and f (s) be a continuous function on K and analytic in the interior of K. Then, for every ε > 0, there exists a polynomial p(s) such that sup We recall that A ∈ B(X) is called a continuity set of the measure P on (X, B(X)) if P(∂A) = 0, where ∂A is a boundary of A. Lemma 12. Let P n , n ∈ N, and P be probability measures on (X, B(X)). Then the following statements are equivalent: 1 • P n converges weakly to P as n → ∞; 2 • For every open set G ⊂ X, lim inf n→∞ P n (G) ≥ P(G); 3 • For every continuity set A of P, lim n→∞ P n (A) = P(A).
The above lemma is a part of Theorem 2.1 from Reference [20]. Now, we can give the proof of Theorem 3. is an open neighbourhood of the element e p 1 (s) , . . . , e p r (s) ∈ S r . Thus, by Theorem 4, P L (G r ε ) > 0, where the distribution P L is defined by (9). Hence, from Theorem 4 again and Lemma 12, lim inf T→∞ P T (G r ε )≥P L (G r ε ) > 0, and the definitions (23) and (32) of P T and G r ε together with (31) prove the first part of the theorem. Second part. Introduce one more set A ε = (g 1 , . . . , g r ) ∈ H r (D) : sup 1≤j≤r sup s∈K j g j (s) − f j (s) < ε .
Then the boundary of A ε lies in the set (g 1 , . . . , g r ) ∈ H r (D) : sup 1≤j≤r sup s∈K j g j (s) − f j (s) = ε , thus, ∂A ε 1 ∩ ∂A ε 2 = ∅ for different ε 1 > 0 and ε 2 > 0. This shows that the set A ε is a continuity set of the measure P L for all but at most countably many ε > 0. Therefore, by Lemma 12, for all but at most countably many ε > 0. Moreover, (31) shows the inclusion G r ε ⊂ A ε . This, (34) and the definitions (23) and (33) of P T and A ε prove the second assertion of the theorem.