# On Zeros of Some Combinations of Dirichlet L-Functions and Hurwitz Zeta-Functions.

1 Introduction

Let s = [sigma] + it be a complex variable, [chi] be a Dirichlet character and [alpha], 0 < [alpha] [less than or equal to] 1, be a fixed parameter. The Dirichlet L-function L(s, [chi]) and Hurwitz zeta-function [zeta](s, [alpha]) are defined, for [sigma] > 1, by the series

[mathematical expression not reproducible]

and continue meromorphically to the whole complex plane. If [chi](m) is a non-principal character, then L(s, [chi]) is an entire function, while if [chi](m) is the principal character modulo q, then L(s, [chi]) has a simple pole at the point s = 1 with residue [[PI].sub.p|q] (1 - 1/p), where p denotes a prime number. The function [zeta](s, [alpha]) has a simple pole at the point s = 1 with residue 1.

It is well known that L(s, [chi]) [not equal to] 0 in the half-plane [sigma] > 1. Moreover, the function L(s, [chi]) has infinitely many zeros lying in the critical strip {s [member of] C : 0 < [sigma] < 1}. The generalized Riemann hypothesis asserts that all these zeros lie on the critical line.

The properties of the function [zeta](s, [alpha]), including the zero-distribution, depend on the arithmetical nature of the parameter [alpha]. In [5], it was proved that the function [zeta](s, [alpha]) with transcendental or rational parameter [alpha] [not equal to] 1, 1/2 has infinitely many zeros in the half-plane {s [member of] C : [sigma] > 1}. J.W.S. Cassels extended [4] the latter result for the case of algebraic irrational [alpha]. The above results also can be found in [14].

It is also known that the function [zeta](s, [alpha]) has zeros in the strip D = {s [member of] C : 1/2 < [sigma] < 1}. Let [alpha] = a/q, (a, q) = 1 and 0 < a < q. Then S.M. Voronin in his thesis [22], see also [23], obtained, that, for every [[sigma].sub.1], [[sigma].sub.2], 1/2 < [[sigma].sub.1] < [[sigma].sub.2] < 1, there exists a constant c = c([alpha], [[sigma].sub.1], [[sigma].sub.2]) > 0 such that, for sufficiently large T, the function [zeta](s, [alpha]) has more than cT zeros lying in the rectangle {s [member of] C : [[sigma].sub.1] < [sigma] < [[sigma].sub.2], [absolute value of t] < T}. S.M. Gonek in [6] proved an analogical result in the case of transcendental [alpha].

The case of algebraic irrational a remains an open problem. The same results on zeros of [zeta](s, [alpha]) in the strip D were obtained independently by B. Bagchi in his thesis [1], they also can be found in [14]. Proofs of the statements on zero distribution of [zeta](s, [alpha]) in D are based on the universality property of [zeta](s, [alpha]). Denote by measA the Lebesgue measure of a measurable set A [subset] R. Then the universality of [zeta](s, [alpha]) is contained in the following statement. Suppose that a is rational [not equal to] 1, 2 or transcendental number. Let K [subset] D be a compact subset with connected complement, and let f (s) be a continuous function on K which is analytic in the interior of K. Then, for every [epsilon] > 0,

[mathematical expression not reproducible].

The later theorem for rational a in non-explicit form was proved by S.M. Voronin [22]. Both the cases of rational and transcendental [alpha] by different methods were treated in [6] and [1].

Clearly, [zeta](s, 1) = [zeta](s) and [zeta](s, 2) = ([2.sup.s] - 1) [zeta](s), where [zeta](s) is the Riemann zeta-function. Therefore, in those cases, [zeta](s, [alpha]) is also universal, however, the approximated function f (s) must be non-vanishing on K. From this, it follows that the universality can not be used to detect zeros of [zeta](s, [alpha]) in D if [alpha] = 1 or [alpha] = 2.

The present paper is devoted to zeros of certain combinations and more general functions of Dirichlet L-functions and Hurwitz zeta-functions.

We say that, for a function L(s), the assertion A([[sigma].sub.1], [[sigma].sub.2]; c, T) is valid if, for every [[sigma].sub.1], [[sigma].sub.2], 1/2 < [[sigma].sub.1] < [[sigma].sub.2] < 1, there exists a constant c > 0 such that, for sufficiently large T, the function L(s) has more than cT zeros lying in the rectangle {s [member of] C : [[sigma].sub.1] < [sigma] < [[sigma].sub.2], 0 < t < T}.

A lot of results on the number of zeros of linear combinations of universal functions can be found in [18]. The paper [19] also is rich by interesting results on zeros of some polynomials of universal functions. For example, a partial case of a corollary of Theorem 2 from [18] is the following theorem.

Theorem 1. Suppose that [[chi].sub.1], ..., [[chi].sub.n] are pairwise non-equivalent Dirichlet characters, and [P.sub.1](s), ..., [P.sub.n](s) are not identically vanishing general Dirichlet series which are absolutely convergent for [sigma] > 1/2, and at least two of the series [P.sub.j] (s) are non-vanishing in the strip D. Then there exists a constant c = c([[chi].sub.1], ..., [[chi].sub.n], [P.sub.1], ..., [P.sub.n], [[sigma].sub.1], [[sigma].sub.2]) > 0 such that, for the linear combination

[P.sub.1](s)L(s, [[chi].sub.1]) + ... + [P.sub.n] (s)L(s, [[chi].sub.n])

the assertion A([[sigma].sub.1], [[sigma].sub.2]; c,T) is valid.

Denote by H(G) the space of analytic functions on G endowed with the topology of uniform convergence on compacta. Let

S = {g [member of] H(D) : 1/g(s) [member of] H(D) or g(s) [equivalent to] 0}

and denote by Un the class of continuous operators F : [H.sup.n](D) [right arrow] H(D) such that, for every open set G [subset] H(D),

([F.sup.-1] G) [intersection] [S.sup.n] [not equal to] [empty set].

Theorem 2. Suppose that [[chi].sub.1],..., [[chi].sub.n] are pairwise non-equivalent Dirichlet characters, and that F [member of] [U.sub.n]. Then there exists a constant c = c([[chi].sub.1], ..., [[chi].sub.n], F, [[sigma].sub.1], [[sigma].sub.2]) > 0 such that, for the function F(L(s, [[chi].sub.1]), ..., L(s, [[chi].sub.n])), the assertion A([[sigma].sub.1], [[sigma].sub.2]; c, T) is valid..

The class [U.sub.n] is theoretical, it is difficult to check its hypotheses. Now we define a simpler class of operators F. Let V be an arbitrary positive number, [D.sub.V] = {s [member of] C : 1/2 < [sigma] < 1, [absolute value of t] < V} and

[mathematical expression not reproducible].

We say that the continuous operator F : [H.sup.n]([D.sub.V]) [right arrow] H([D.sub.V]) belongs to the class [U.sub.n,V] if, for every polynomial p = p(s),

([F.sup.-1]{p}) [intersection] [S.sup.n.sub.V] [not equal to] [empty set].

Theorem 3. Suppose that [[chi].sub.1],..., [[chi].sub.n] are pairwise non-equivalent Dirichlet characters, and that F [member of] [U.sub.n,V] with sufficiently large V. Then there exists a constant c = c([[chi].sub.1], ..., [[chi].sub.n], F, [[sigma].sub.1], [[sigma].sub.2]) > 0 such that, for the function F(L(s, [[chi].sub.1]), ..., L(s, [[chi].sub.n])) the assertion A([[sigma].sub.1], [[sigma].sub.2]; c, T) is valid with T < V.

We give an example of F [member of] [U.sub.V]. Let F([g.sub.1],[g.sub.2]) = [g.sup.2.sub.1] + [g.sup.2.sub.2], [g.sub.1], [g.sub.2] [member of] H([D.sub.V]). Let p(s) be an arbitrary polynomial. Then there exists a constant [c.sub.1] > 0 such that [absolute value of p(s)] [less than or equal to] [c.sub.1] for s [member of] [D.sub.V]. We take C > [c.sub.1], and

[p.sub.1](s) = p(s) + C/2 [square root of C], [p.sub.2](s) - C/2i [square root of C].

Then we have that [p.sub.1](s) [not equal to] 0 and [p.sub.2](s) [not equal to] 0 on [D.sub.V]. Moreover,

[p.sup.2.sub.1] (s) + [p.sup.2.sub.2] (s) = p(s).

Thus, ([p.sub.1], [p.sub.2]) [member of] ([F.sup.-1]{p}) [intersection] [S.sup.2.sub.V]. Therefore, if [[chi].sub.1] and [[chi].sub.2] are non-equivalent characters, then the function

[L.sup.2](s, [[chi].sub.1]) + [L.sup.2](s, [[chi].sub.2])

has more than cT zeros ir the rectangle {s [member of] C : [[sigma].sub.1] < [sigma] < [[sigma].sub.2], 0 < t < T} with sufficiently large T < V.

The second example is of the following form. Let [P.sub.1](s),..., [P.sub.n](s) be analytic functions on D, two of them are non-vanishing, and the remaining functions are bounded by polynomials. Then, for the function

[P.sub.1](s)L(s, [[chi].sub.1]) + ... + [P.sub.n](s)L(s, [[chi].sub.n]),

where [[chi].sub.1],..., [[chi].sub.n] are pairwise non-equivalent Dirichlet characters, the assertion A([[sigma].sub.1], [[sigma].sub.2]; c, T) is valid. Indeed, the operator F : [H.sup.n]([D.sub.V]) [right arrow] H([D.sub.V]) given by the formula

F([g.sub.1], ..., [g.sub.n]) = [P.sub.1][g.sub.1] + ... + [P.sub.n][g.sub.n], [g.sub.1], ..., [g.sub.n] [member of] H([D.sub.V]),

is continuous. Let p = p(s) be an arbitrary polynomial. Without loss of generality, we can suppose that [P.sub.1](s) and [P.sub.2](s) are non-vanishing for s [member of] D. Moreover, there exist polynomials [q.sub.j](s), j = 3, ..., n, such that, for s [member of] D, [absolute value of [P.sub.j](s)] [less than or equal to] [absolute value of [q.sub.j](s)], j = 3, ..., n. We take

[mathematical expression not reproducible],

where C > 0 is such that p(s) + C [not equal to] 0 and -([P.sub.3](s) + ... + [P.sub.n](s)) - C [not equal to] 0 on [D.sub.V]. Then we have that

[P.sub.1][g.sub.1] + ... + [P.sub.n] [g.sub.n] = P

and [g.sub.1], ..., [g.sub.n] [not equal to] 0 on [D.sub.V]. Therefore the hypotheses of Theorem 3 are satisfied.

Now we state the results on zeros of certain functions related to Hurwitz zeta-functions. Define

L ([[alpha].sub.1], ..., [[alpha].sub.n]) = {log(m + [[alpha].sub.j]) : m [member of] [N.sub.0] = N [union] {0}, j = 1, ..., n}.

Theorem 4. Suppose that the set L([[alpha].sub.1], ..., [[alpha].sub.n]) is linearly independent over the field of rational numbers Q, and [c.sub.1], ..., [c.sub.n] are complex numbers, at least one of them is non-zero. Then there exists a constant c = c([[alpha].sub.1], ..., [[alpha].sub.n], [c.sub.1], ..., [c.sub.n], [[alpha].sub.1], [[alpha].sub.2]) > 0 such that, for the function

[c.sub.1] [zeta](s, [[alpha].sub.1]) + ... + [c.sub.n][zeta](s, [[alpha].sub.n])

the assertion A([[sigma].sub.1], [[sigma].sub.2]; c, T) is valid..

Denote by [[??].sub.n] the class of continuous operators F : [H.sup.n](D) [right arrow] H(D) such that, for every open set G [subset] H(D), the set [F.sup.-1] G is non-empty.

Theorem 5. Suppose that the set L([[alpha].sub.1], ..., [[alpha].sub.n]) is linearly independent over Q, and that F [member of] [[??].sub.n]. Then there exists a constant c = c([[alpha].sub.1], ..., [[alpha].sub.n], F, [[sigma].sub.1], [[sigma].sub.2]) > 0 such that, for the function F ([zeta](s, [[alpha].sub.1]), ...,[zeta] (s, [[alpha].sub.n])), the assertion A([[sigma].sub.1], [[sigma].sub.2]; c, T) is valid.

Now we consider a simpler class of operators. Denote by [[??].sub.n1] the class of continuous operators F : [H.sup.n](D) [right arrow] H(D) such that each polynomial p(s) has its preimage [F.sup.-1]{p}.

Theorem 6. Suppose that the set L([[alpha].sub.1], ..., [[alpha].sub.n]) is linearly independent over Q, and that F [member of] [[??].sub.n1]. Then there exists a constant c = c([[alpha].sub.1], ..., [[alpha].sub.n], F, [[sigma].sub.1], [[sigma].sub.2]) > 0 such that, for the function F([zeta](s, [[alpha].sub.1]), ..., [zeta](s, [[alpha].sub.n])), the assertion A([[sigma].sub.1], [[sigma].sub.2]; c, T) is valid.

For example, the operator F : [H.sup.2](D) [right arrow] H(D) defined by the formula F([g.sub.1], [g.sub.2]) = [g.sub.2] + [g.sub.2] belongs to the class [[??].sub.21]. Really, for any polynomial p(s), there exist two polynomials

[p.sub.1](s) = [p(s) + 1]/2 and [p.sub.2](s) = [p(s) - 1]/2i,

and [p.sup.2.sub.1] (s) + [p.sup.2.sub.2] (s) = p(s).

Remark 1. In view of Theorem 6, the numbers [c.sub.1], ..., [c.sub.n] in Theorem 4 can be replaced by functions analytic in D.

Now we consider a collection of functions L(s, [[chi].sub.1]), ..., L(s, [[chi].sub.l]), [zeta](s, [[alpha].sub.1]), ..., [zeta](s, [[alpha].sub.r]). Denote by P the set of all prime numbers, and define the set

L (P, [[alpha].sub.1], ..., [[alpha].sub.r]) = {(log p : p [member of] P), (log(m + [[alpha].sub.j]) : m [member of] [N.sub.0], j = 1, ..., r)}.

Theorem 7. Suppose that [[chi].sub.1], ..., [[chi].sub.l] are pairwise non-equivalent Dirichlet characters, and that the set L (P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q. Let [mathematical expression not reproducible] be complex numbers, at least one of [[??].sub.j] is non-zero. Then there exists a positive constant [mathematical expression not reproducible] such that, for the function

[mathematical expression not reproducible]

the assertion A([[sigma].sub.1], [[sigma].sub.2]; c, T) is valid..

We say that a continuous operator F : [H.sup.l+r](D) [right arrow] H(D) belongs to the class [U.sub.l,r] if, for every polynomial p(s), the set ([F.sup.-1]{p}) [intersection] ([S.sup.l] x [H.sup.r](D)) is non-empty.

Theorem 8. Suppose that [[chi].sub.1], ..., [[chi].sub.l] are pairwise non-equivalent Dirichlet characters, the set L (P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q, and that F [member of] [U.sub.l,r]. Then there exists a constant c = c([[chi].sub.1], ..., [[chi].sub.l], [[alpha].sub.1], ..., [[alpha].sub.r], F, [[sigma].sub.1], [[sigma].sub.2]) > 0 such that, for the function

F (L(s, [[chi].sub.1]), ... L(s, [[chi].sub.l]), [zeta](s, [[alpha].sub.1]) ..., [zeta](s, [[alpha].sub.r]))

the assertion A([[sigma].sub.1], [[sigma].sub.2]; c, T) is valid..

For example, if [[chi].sub.1] and [[chi].sub.2] are non-equivalent characters, and the numbers [[alpha].sub.1] and [[alpha].sub.2] are algebraically independent over Q, then the function L(s, [[chi].sub.1]) L(s, [[chi].sub.2]) [zeta](s, [[alpha].sub.1]) [zeta](s, [[alpha].sub.2]) satisfies the hypotheses of Theorem 8, since a polynomial p(s) has a preimage (1,1,1,p(s)) [member of] [S.sup.2] x [H.sup.2](D), and the algebraic independence of the numbers [[alpha].sub.1] and [[alpha].sub.2] implies the linear independence of the set L(P, [[alpha].sub.1], [[alpha].sub.2]).

Remark 2. In view of Theorem 8, the numbers [c.sub.j] and [[??].sub.j] in Theorem 7 can be replaced by functions analytic in D.

2 Universality

In Introduction, we have mentioned the universality of the Hurwitz zeta-function. In this section, we present more general universality results which will be applied for the proof of Theorems 1-8. We start with a joint universality theorem for Dirichlet L-functions. In this theorem, a collection of shifts of Dirichlet L-functions with non-equivalent Dirichlet characters, uniformly on compact sets of the strip D, can approximate simultaneously a collection of non-vanishing analytic functions.

Theorem 9. Suppose that [[chi].sub.1],..., [[chi].sub.n] are pairwise non-equivalent Dirichlet characters. For j = 1, ..., n, let [K.sub.j] be a compact set of the strip D with connected complement, and let [f.sub.j] (s) be a continuous non-vanishing function on [K.sub.j] which is analytic in the interior of [K.sub.j]. Then, for every [epsilon] > 0,

[mathematical expression not reproducible].

Theorem 9 in a weaker non-explicit form has been obtained by S.M. Voronin in [21], and applied for the functional independence of Dirichlet L-functions. First in a weaker explicit form the theorem was given in [22], see also [23]. Similar results independently by different methods were obtained by S.M. Gonek [6] and B. Bagchi [1,2]. In the present form, the theorem was stated in [20] and discussed in [11]. A full proof of Theorem 9 is given in the master work [17].

Theorem 10. Suppose that [[chi].sub.1],..., [[chi].sub.n] are pairwise non-equivalent Dirichlet characters, and that F [member of] [U.sub.n]. Let K [subset] D be a compact set with connected complement, and let f(s) be a continuous function on K which is analytic in the interior of K. Then, for every [epsilon] > 0,

[mathematical expression not reproducible].

The theorem is proved in [11], Theorem 4.

Theorem 11. Suppose that [[chi].sub.1], ..., [[chi].sub.n are pairwise non-equivalent Dirichlet characters, and that K and f (s) are the same as in Theorem 10. Let V > 0 be such that K [subset] [D.sub.V], and that F [member of] [U.sub.n,V]. Then the same assertion as in Theorem 10 is valid.

The proof of the theorem is given in [11], Theorem 5.

Now we state joint universality theorems for Hurwitz zeta-functions.

Theorem 12. Suppose that the set L([[alpha].sub.1], ..., [[alpha].sub.n]) is linearly independent over Q. For j = 1, ..., n, let [K.sub.j] be a compact subset of the strip D with connected complement, and let [f.sub.j] (s) be a continuous function on [K.sub.j] which is analytic in the interior of [K.sub.j]. Then, for every [epsilon] > 0,

[mathematical expression not reproducible].

The theorem is proved in [9].

Theorem 12 implies the following two universality theorems for composite functions obtained in [12].

Theorem 13. Suppose that the set L([[alpha].sub.1], ..., [[alpha].sub.n]) is linearly independent over Q, and that F [member of] [[??].sub.n1]. Let K and f (s) be the same as in Theorem 10. Then, for every [epsilon] > 0,

[mathematical expression not reproducible].

Theorem 14. Suppose that the set L([[alpha].sub.1], ..., [[alpha].sub.n]) is linearly independent over Q, and that F [member of] [[??].sub.n1]. Let K and f (s) be the same as in Theorem 10. Then the same assertion as in Theorem 13 is true.

In [12], theorems on universality of F ([zeta](s, [[alpha].sub.1]), ..., [zeta](s, [[alpha].sub.n])) for some other classes of operators F also can be found.

Denote by B(S) the class of Borel sets of the space S, and on ([H.sup.l+r] (D), B([H.sup.l+r] (D))) define the probability measure [P.sub.T] by

[mathematical expression not reproducible].

To state a limit theorem for the measure [P.sub.T], we need some notation and definitions. Denote by [gamma] the unit circle on the complex plane, and define

[mathematical expression not reproducible],

where [[gamma].sub.p] = [gamma] for all primes p, and [[gamma].sub.m] = [gamma] for all m [member of] [N.sub.0]. The [??] and [OMEGA] with the product topology and pointwise multiplication are compact topological Abelian groups. Let [[OMEGA].bar] = [??] x [[OMEGA].sub.1] x ... x [[OMEGA].sub.r], where [[OMEGA].sub.j] = [OMEGA] for j = 1, ..., r. Then again [[OMEGA].bar] is a compact topological Abelian group. Therefore, on ([[OMEGA].bar], B([[OMEGA].bar])), the probability Haar measure [m.sub.H] can be defined, and this leads to the probability space ([[OMEGA].bar], B([[OMEGA].bar]), [m.sub.H]). Note that the measure [m.sub.H] is the product of the Haar measures [mathematical expression not reproducible], respectively, j = 1, ..., r. Denote elements of [mathematical expression not reproducible]. Moreover, let [mathematical expression not reproducible]. Now, on the probability space ([[OMEGA].bar], B([[OMEGA].bar]),[m.sub.H]), define the [H.sup.l+r](D) valued random element [XI](s, [[omega].bar]) by the formula

[mathematical expression not reproducible],

where

[mathematical expression not reproducible],

and

[mathematical expression not reproducible].

Let [P.sub.[XI]] be the distribution of the random element [XI](s, [[omega].bar]), i.e., for A [member of] B([H.sup.l+r] (D)),

[mathematical expression not reproducible].

Proposition 1. Suppose that the set L(P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q. Then the measure Pt converges weakly to [P.sub.[XI]] as T [right arrow] [infinity].

Proof of the proposition is based on the following lemmas.

Lemma 1. Suppose that the set L(P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q. Then the probability measure

[mathematical expression not reproducible],

converges weakly to the Haar measure as T [right arrow] [infinity].

Proof. As in the proof of Theorem 3 in [10], we have that the Fourier transform [mathematical expression not reproducible] of the measure [Q.sub.T] is given by

[mathematical expression not reproducible], (2.1)

where only a finite number of integers [k.sub.p] and [l.sub.jm], j = 1, ..., r, are distinct from zero. Now we use essentially the fact that the set L(P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q, and, after integration, we obtain from (2.1) that

[mathematical expression not reproducible].

This proves the lemma.

Let [[sigma].sub.1] > 1/2 be a fixed number, and

[mathematical expression not reproducible].

Define the series

[mathematical expression not reproducible].

It follows by a standard way that the series for [mathematical expression not reproducible] converge absolutely for [[sigma].sub.1] > 1/2. Define the probability measures

[mathematical expression not reproducible].

Lemma 1, the absolute convergence of the above series and a property of the weak convergence of probability measures related by continuous mappings imply the following lemma.

Lemma 2. Suppose that the set L(P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q. Then, on ([H.sup.l+r] (D), B([H.sup.l+r] (D))), there exists a probability measure [P.sub.n] such that both the measures [P.sub.T,n] and [[??].sub.T,n] converges weakly to [P.sub.n] as T [right arrow] [infinity].

The next step of the proof of Proposition 1 consists of the approximation of the functions L(s, [[chi].sub.j]) and [zeta](s, [[alpha].sub.j]) by [L.sub.n](s, [[chi].sub.j]) and [[zeta].sub.n](s, [[alpha].sub.j]), respectively. Let [rho] be the metric on H(D) inducing the topology of uniform convergence on compacta, see, for example, [8]. Define the metric on [H.sup.l+r](D) by

[mathematical expression not reproducible]

where [mathematical expression not reproducible]. Using the known onedimensional results of [7] and [15] for periodic zeta-functions, we obtain approximations in the space [H.sup.l+r](D).

Lemma 3. We have

[mathematical expression not reproducible].

In the case of [zeta](s, [[alpha].sub.j], [[omega].sub.j]), a one-dimensional approximation is obtained for transcendental [[alpha].sub.j], however, the transcendence is used only for a linear independence of the set {log(m + [[alpha].sub.j]) : m [member of] [N.sub.0]}. Therefore, we have the following analogue of Lemma 3.

Lemma 4. Suppose that the set L(P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q. Then, for almost all [[omega].bar] [member of] [[OMEGA].bar].

[mathematical expression not reproducible].

Define one more probability measure

[mathematical expression not reproducible].

Lemma 5. Suppose that the set L(P, [[alpha].sujb.1], ..., [[alpha].sub.r]) is linearly independent over Q. Then, on ([H.sup.l+r](D), B([H.sup.l+r] (D))), there exists a probability measure P such that both the measures [P.sub.T] and [[??].sub.T] converges weakly to P as T [right arrow] [infinity].

The lemma is a consequence of Lemmas 2-4. Its proof runs similarly to that of Theorem 6 from [10].

For the proof of Proposition 1, it is sufficient to show that the measure P in Lemma 5 coincides with [P.sub.[XI]]. For this, again the linear independence of the set L(P, [[alpha].sub.1],..., [[alpha].sub.r]) is applied.

Let [mathematical expression not reproducible]. Define a family {[[PHI].sub.[tau]] : [tau] [member of] R} of measurable measure preserving transformations on [[OMEGA].bar] by the formula [mathematical expression not reproducible]. This family is a one-parameter group. A set A [member of] B([[OMEGA].bar]) is called invariant with respect to the group {[[PHI].sub.[tau]] : t [member of] R} if, for each t [member of] R, the sets A and [A.sub.[tau]] = [[PHI].sub.[tau]] (A) may differ one from another only by [m.sub.H]-measure zero. All invariant sets form a [sigma]-subfield of B([[OMEGA].bar]). The group {[[PHI].sub.[tau]] : [tau] [member of] R} is ergodic, if the latter [sigma]-subfield consists only of the sets of [m.sub.H]-measure zero or one.

Lemma 6. Suppose that the set L(P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q. Then the group {[[PHI].sub.[tau]] : [tau] [member of] R} is ergodic.

Proof. A method of Fourier transform is used, and the arguments are similar to those of Lemma 7 from [10], where the case of the algebraically independent numbers [[alpha].sub.1], ..., [[alpha].sub.r] was considered.

Proof of Proposition 1. On the probability space ([[OMEGA].bar], B([[OMEGA].bar]), [m.sub.H]), define a random variable [xi] by the formula

[mathematical expression not reproducible],

where A is a fixed continuity set of the measure P in Lemma 5. Thus

[mathematical expression not reproducible].

Lemma 6 implies the ergodicity of the random process [xi]([[PHI].sub.[tau]]([[omega].bar])). Therefore, by the Birkhoff-Khintchine theorem,

[mathematical expression not reproducible],

where E[xi] is the expectation of [xi]. On the other hand, the definitions of [xi] and [[PHI].sub.[tau]] yield the equalities

[mathematical expression not reproducible].

Combining the above four equalities, we find that P(A) = [P.sub.[XI]](A). Since A was arbitrary, this is true for all continuity sets of the measure P, hence, P(A) = [P.sub.[XI]] (A) for all A [member of] B([H.sup.l+r] (D)). The proposition is proved.

For the proof of universality theorems, we additionally need the support of the measure [P.sub.[XI]]. Since the space [H.sup.l+r] (D) is separable, the support of [P.sub.[XI]] is a minimal closed set [S.sub.[XI]] [member of] B([H.sup.l+r](D)) such that [P.sub.[XI]] ([S.sub.[XI]]) = 1.

Proposition 2. Suppose that [[chi].sub.1], ..., [[chi].sub.l] are pairwise non-equivalent Dirichlet characters, and that the set L(P, [[alpha].sub.1],..., [[alpha].sub.r]) is linearly independent over Q. Then the support of the measure [P.sub.[XI]] is the set [S.sup.l] x [H.sup.r] (D).

Proof. Since the characters [[chi].sub.1],..., [[chi].sub.l] are pairwise non-equivalent, the support of the random element

[mathematical expression not reproducible]

is the set [S.sup.l] [11]. The linear independence of the set L(P, [[alpha].sub.1],..., [[alpha].sub.r]) implies that of the set L([[alpha].sub.1],..., [[alpha].sub.r]). Therefore, by [9], the support of the random element [mathematical expression not reproducible]. Since the spaces [H.sup.l] (D) and [H.sup.r] (D) are separable, we have [3] that

B([H.sup.l+r](D)) = B([H.sup.l](D)) x B(Hr(D)). (2.2)

Let [m.sup.r.sub.H] be the Haar measure on [mathematical expression not reproducible]. Then [m.sub.H] is the product of the measures [[??].sub.h] and [m.sup.r.sub.H]. Let [A.sb.1] [member of] B([H.sup.l] (D)) and [A.sub.2] [member of] B([H.sup.r] (D)). Then, in view of (2.2), it suffices to consider [P.sub.[XI]](A) with A = [A.sub.1] x [A.sub.2]. Thus, by the above remark,

[mathematical expression not reproducible].

This, together with supports of the random elements [mathematical expression not reproducible] proves the proposition.

Now we are ready to prove a joint universality theorem for a collection L(s, [[chi].sub.1], ..., L(s [[chi].sub.l]), [zeta](s, [[alpha].sub.1]), ..., [zeta](s, [[alpha].sub.r]).

Theorem 15. Suppose that [[chi].sub.1],..., [[chi].sub.l] are pairwise non-equivalent Dirichlet characters, and that the set L(P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q. For j = 1, ..., l, let [K.sub.j] [subset] D be a compact subset with connected complement, and let [f.sub.j](s) be a continuous non-vanishing function on [K.sub.j] which is analytic in the interior of [K.sub.j]. For j = 1, ..., r, let [[??].sub.j] [subset] D be a compact subset with connected complement, and let [[??].sub.j] (s) be a continuous function on [[??].sub.j] which is analytic in the interior of [[??].sub.j]. Then, for every [epsilon] > 0,

[mathematical expression not reproducible].

Proof. By the Mergelyan theorem [16], see also [24], there exist polynomials [p.sub.j](s), j = 1, ..., l, and [q.sub.j](s), j = 1,..., r, such that

[mathematical expression not reproducible], (2.3)

[mathematical expression not reproducible]. (2.4)

Define the set

[mathematical expression not reproducible].

Then, in view of Proposition 2, G is an open neighbourhood of the element [mathematical expression not reproducible] of the support of the measure [P.sub.[XI]]. Thus, [P.sub.[XI]] (G) > 0. Therefore, by Proposition 1,

[mathematical expression not reproducible].

From this, the definition of G and equalities (2.3) and (2.4), the theorem follows.

Proposition 3. Suppose that the set L(P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q, and that F [member of] [U.sub.l,r]. Then the probability measure

[mathematical expression not reproducible],

converges weakly to [P.sub.[XI]] [F.sup.-1] as T [right arrow] [infinity].

Proof. The proposition is a consequence of Proposition 1, of the continuity of F and Theorem 5.1 from [3]. []

Proposition 4. Suppose that [[chi].sub.1], ..., [[chi].sub.l] are pairwise non-equivalent Dirichlet characters, the set L(P, [[alpha].sub.1], ..., [[alpha].sub.r]) is linearly independent over Q, and that F [member of] [U.sub.l,r]. Then the support of the measure [P.sub.[XI]] [F.sup.-1] is the whole of H(D).

Proof. Let g be an arbitrary element of H(D), and G be its open neighbourhood. From the continuity of F, we have that the [F.sup.-1]G is open, too.

It is well known that the approximation in the space H(D) reduces to an approximation on compact subsets with connected complement, see, for example, [13]. Therefore, by the Mergelyan theorem, there exists a polynomial p = p(s) such that p [member of] G. Since, by the hypothesis of the proposition, the set ([F.sup.-1]{p}) [intersection] ([S.sup.l] x [H.sup.r](D)) is non-empty, we have that the set ([F.sup.-1]G) [intersection] ([S.sup.l] x [H.sup.r](D)) also is non-empty. This means that the set [F.sup.-1]G is an open neighbourhood of the element [g.bar] [member of] [S.sup.l] x [H.sup.r](D). In view of Proposition 2, the set [S.sup.l] x [H.sup.r] (D) is the support of the measure [P.sub.[XI]], thus [P.sub.[XI]] ([F.sup.-1]G) > 0. Hence,

[P.sub.[XI]] [F.sup.-1](G) = [P.sub.[XI]] ([F.sup.-1]G) > 0.

Since g and G are arbitrary, this proves the proposition.

Theorem 16. Suppose that [[chi].sub.1],..., [[chi].sub.l] are pairwise non-equivalent Dirichlet characters, and that the set L(P, [[alpha].sub.1],..., [[alpha].sub.r]) is linearly independent over Q, and that F [member of] [U.sub.lr]. Let K [subset] D 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 [epsilon] > 0,

[mathematical expression not reproducible].

Proof. We repeat the arguments similar to those of the proof of Theorem 15.

By the Mergelyan theorem, there exists a polynomial p(s) such that

[mathematical expression not reproducible]. (2.5)

Define the set

[mathematical expression not reproducible].

Then, by Proposition 4, the set G is an open neighbourhood of the element p(s) of the support of [P.sub.[XI]][F.sup.-1]. Therefore, Proposition 3 implies the inequality

[mathematical expression not reproducible].

Combining this with (2.5) and the definition of G completes the proof.

3 Proof of the main theorems

Proofs of all theorems on zeros are based on the corresponding universality theorems as well as on the classical Rouche theorem.

Proof of Theorem 2. We take f (s) = s - [[sigma].sub.0] in Theorem 10. Then, by Theorem 10, for every [epsilon] > 0, the set of [tau] [member of] R satisfying the inequality

[mathematical expression not reproducible]

has a positive lower density. Now we take e such that

[mathematical expression not reproducible].

Thus, the functions [mathematical expression not reproducible] satisfy the hypotheses of the Rouche theorem. This proves the theorem.

Proof of Theorem 3. We use Theorem 11 and repeat the proof of Theorem 2.

Proof of Theorem 4. We use the notation

[mathematical expression not reproducible].

Without loss of generality, we may suppose that [c.sub.1] [not equal to] 0. Let

[f.sub.1](s) = [c.sup.-1.sub.1] (s - [[sigma].sub.0]), [f.sub.2](s) = ... = [f.sub.n](s) = [epsilon] > 0,

where the number e satisfies inequality

[mathematical expression not reproducible].

Suppose that reals [tau] satisfy the inequality

[mathematical expression not reproducible]

From this, we find that

[mathematical expression not reproducible].

Therefore, by the definition of [f.sub.1](s), ..., [f.sub.n](s),

[mathematical expression not reproducible]

and the proof finishes by application of Theorem 12 and the Rouche theorem.

Proof of Theorem 5. We use Theorem 13 and the Rouche theorem, and argue similarly to the proof of Theorem 2.

Proof of Theorem 6. The proof coincides with that of Theorem 5. In place of Theorem 13, we apply Theorem 14.

Proof of Theorem 7. Without loss of generality, we may assume that ci = 0. We take in Theorem 15

[mathematical expression not reproducible],

where the positive number e satisfies the inequality

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Let [tau] satisfy the inequalities

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Then, for these [tau], we find that

[mathematical expression not reproducible].

Moreover,

[mathematical expression not reproducible].

Two latter inequalities show that

[mathematical expression not reproducible].

From this and the definition of [epsilon], it follows that the functions

[mathematical expression not reproducible]

and s - [[sigma].sub.0] in the disc [absolute value of (s - [[sigma].sub.0])] [less than or equal to] [[rho].sub.0] satisfy the hypotheses of the Rouche theorem. Hence, the theorem follows.

Proof of Theorem 8. We use Theorem 16 and repeat the proof, for example, of Theorem 2.

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https://doi.org/10.3846/13926292.2017.1365313

Virginija Garbaliauskiene (a), Julija Karaliunaite (b) and Antanas Laurincikas (c)

(a) Faculty of Technology, Physical and Biomedical Sciences, Siauliai University Vilniaus str. 141, LT-76353 Siauliai, Lithuania

(b) Faculty of Fundamental Sciences, Vilnius Gediminas Technical University Sauletekio al. 11, LT-10223 Vilnius, Lithuania

(c) Faculty of Mathematics and Informatics, Vilnius University Naugarduko str. 24, LT-03225 Vilnius, Lithuania

E-mail (corresp.): vgarbaliauskiene@gmail.com

E-mail: julija.karaliunaite@vgtu.lt

Received June 4, 2017; revised August 3, 2017; published online November 15, 2017
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