VILNIUS UNIVERSITY. Santa Ra kauskiene JOINT UNIVERSALITY OF ZETA-FUNCTIONS WITH PERIODIC COEFFICIENTS

Transcription

1 VILNIUS UNIVERSITY Santa Ra kauskiene JOINT UNIVERSALITY OF ZETA-FUNCTIONS WITH PERIODIC COEFFICIENTS Doctoral dissertation Physical sciences, mathematics 0P) Vilnius, 202

2 The work on this dissertation was performed at the Institute of Mathematics and Informatics of Vilnius University in Scientic supervisor: Prof. Dr. Habil. Antanas Laurin ikas Vilnius University, Physical sciences, Mathematics - 0P)

3 VILNIAUS UNIVERSITETAS Santa Ra kauskiene DZETA FUNKCIJU SU PERIODINIAIS KOEFICIENTAIS JUNGTINIS UNIVERSALUMAS Daktaro disertacija Fiziniai mokslai, matematika 0P) Vilnius, 202

4 Disertacija rengta metais Vilniaus universiteto Matematikos ir Informatikos Institute. Mokslinis vadovas: Prof. habil. dr. Antanas Laurin ikas Vilniaus universitetas, ziniai mokslai, matematika - 0P)

5 Contents Introduction 7 Actuality Aims and problems Methods Novelty History of the problem Approbation Principal publications Acknowledgment Joint universality for periodic Hurwitz zeta-functions 20.. Statement of the main theorem Joint limit theorem Support of the limit measure Proof of Theorem Extended joint universality theorem for periodic Hurwitz zeta-functions Statement of an extended joint universality theorem Extended joint limit theorem Support of the limit measure Proof of Theorem Mixed joint universality for periodic Hurwitz zeta-functions and the Riemana zetafunction Statement of a mixed joint universality theorem Joint limit theorem for periodic Hurwitz zeta-functions and the Riemann zeta-function Support of the limit measure Proof of Theorem

6 4 Mixed joint universality for periodic Hurwitz zeta-functions and the zeta-function of cusp form Statement of the main theorem Joint limit theorem for periodic Hurwitz zeta-functions and the function ζs, F ) Support of the limit measure Proof of Theorem Conclusions 68 Bibliography 69 Notation 72 6

7 Introduction In the thesis, the joint universality of periodic Hurwitz zeta-functions as well as that jointly with the Riemann zeta-function or zeta-functions of normalized cusp forms is obtained. Actuality Universality is a very important and useful property of zeta and L-functions, it has a series of theoretical and practical applications. Universality is the main ingredient in the proof of the functional independence of zeta and L-functions, is applied in the investigation of zero-distribution and moment problem, allows to prove various value denseness theorems, and, of course, plays a crucial role in approximation of analytic functions. One of possible practical applications is estimation of integrals over complicated analytic curves in quantum mechanics [4]. Thus, this is a motivation to extend the class of universal functions. In practice, often approximation and estimation of systems of analytic functions is needed. This problem can be successfully solved using the joint universality of zeta-functions. The majority of zeta and L-functions have approximate functional equations, therefore, due to joint universality, simultaneous estimation of analytic functions reduces to that of rather simple Dirichlet polynomials. This is a singnicant impact of the universality of zeta-functions to the theory of analytic functions. After Voronin's remarkable work [42], a series of famous number theorists continued their investigations on the universality of zeta-functions. The names of B. Bagchi, H. Bauer, R. Garunk²tis, P. Gauthier, S. M. Gonek, J. Kaczorowski, A. Laurin ikas, K. Matsumoto, A. Reich, J. Steuding, the works of young Lithuanian, Japanese, German and Polish mathematicians clearly show the actuality of the universality problem in the theory of zeta and L-functions. Aims and problems The aim of the thesis is to extend the joint universality to new classes of zeta-functions. The concrete problems are the following.. To remove a rank condition in a joint universality theorem for periodic Hurwitz zeta-functions. 7

8 2. To weaken a rank condition in an extended joint universality theorem a collection of periodic sequences corresponds each shift parameter) for periodic Hurwitz zeta-functions. 3. To prove a mixed joint universality theorem for the Riemann zeta-function and periodic Hurwitz zeta-functions. 4. To prove a mixed joint universality theorem for a zeta-function of normalized Hecke eigen cusp forms and periodic Hurwitz zeta-functions. Methods In the thesis, for the proof of joint universality theorems for zeta-functions an analytic method based on probabilistic limit theorems on the weak convergence of probability measures in the space of analytic functions is applied. This method also involves elements of the measure theory and the approximation theory of analytic functions. Novelty All results of thesis are new. They improve or extend joint universality results for periodic Hurwitz zeta-functions. History of the problem In 975, S. M. Voronin discovered [42] the universality of the Riemann zeta-function ζs), s = σ + it. Roughly speaking, he proved that any non-vanishing analytic function can be approximated uniformly on some sets of the strip D = s C : 2 < σ < by shifts ζs + iτ), τ R. We state a modern version of the Voronin theorem which proof is given in [9]. measa denotes the Lebesgue measure of a measurable set A R. Theorem A. Suppose that K D is a compact subset with connected complement, and that fs) is a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup ζs + iτ) fs) < ε s K > 0. Theorem A shows that the set of shifts ζs+iτ) approximating a given analytic function is innite: it has a positive lower density. A proof of Theorem A is dierent from the initial Voronin proof, and is based on a limit theorem on the weak convergence of probability measures in the space of analytic 8

9 functions. The latter method was proposed by B. Bagchi in this thesis [], and was developed in the monographs [9], [27] and [4]. It turned out that some other zeta-functions also have the universality property. The zeta-functions of cusp forms are among universal in the Voronin sense functions. We remind that the function F s) is callied a cusp form of weight κ with respect to the full modular group SL2, Z) = a b : a, b, c, d Z, ad bc = c d if is holomorphic in the upper half-plane Imz > 0, with some κ 2N satises, for all a c SL2, Z), the functional equation ) az + b F = cz + d) κ F z), cz + d and at innity has the Fourier series expansion F z) = cm)e 2πimz. m= Moreover, we assume that the cusp form F s) is a simultaneous eigen function of all Hecke operators T n f)z) = n κ d n ) nz + bd f, n N. dz d κ d b=0 It is known that, in this case, c) 0. Thus, we can normalize the function F s) by taking c) =. To a normalized Hecke eigen cusp form F z), we can attach the zeta-function ζs, F ) dened, for σ > κ+ 2, by ζs, F ) = m= cm) m s = p αp) ) p s βp) ) p s, where, for primes p, αp) and βp) are conjugate complex numbers such that αp) + βp) = cp). It is well known that the function ζs, F ) has analytic continuation to an entire function. The theory of modular forms is given, for example, in [] and [7]. The universality of the function ζs, F ) was began to study in [5] and completely proved in [29]. κ Let D κ = s C : 2 < σ < κ+ 2. Then the following analogue of Theorem A is true. Theorem B. Suppose that K D κ is a compact subset with connected complement, and that fs) is a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup ζs + iτ, F ) fs) < ε s K > 0. b d 9

10 A more interesting and complicated property of zeta-functions than the universality is their joint universality. The rst result on joint universality also belongs to S. M. Voronin. In [43], see also [8], he obtained a joint universality theorem for Dirichlet L-functions. We remind that two Dirichlet characters χ and χ 2 are equivalent if they are generated by the same primitive character. The theory of Dirichlet L-functions can be found, for example, in [38], [7]. We state a modied version of the Voronin theorem. Theorem C. Suppose that χ,..., χ r are pairwise non-equivalent Dirichlet characters, and Ls, χ ),..., Ls, χ r ) are the corresponding Dirichlet L-functions. For j =,..., r, let K j D be a compact subset with connected complement, and let f j s) be a continuous non-vanishing function on K j which is analytic in the interior of K j. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup j r sup Ls + iτ, χ j ) f j s) < ε s K j > 0. Other versions of Theorem C were independently obtained by S. M. Gonek [8] and B. Bagchi [], [2]. The Voronin theorem in the form of Theorem C is given in [26]. In Theorem C, a collection of analytic functions are simultaneously approximated by shifts of Dirichlet L-functions. This procedure, of course, requires a certain independence of a collection of L-functions, and this independence is expressed by the non-equivalence of Dirichlet characters. The known joint universality theorems for other zeta-functions also involve some independence hypotheses. This is clearly reected in a joint universality theorem for Hurwitz zeta-functions. However, rst we remind the denition and universality of the Hurwitz zeta function. Let α, 0 < α, be a xed parameter. The Hurwitz zeta-function ζs, α) is dened, for σ >, by the series ζs, α) = m=0 m + α) s, and has meromorphic continuation to the whole complex plane with unique simple pole at the point s = with residue. The function ζs, α) is an interesting analytical object depending on a parameter α whose arithmetical nature inuences the properties of ζs, α). The universality of ζs, α) is contained in the following theorem. Theorem D. Suppose that the number α is transcendental or rational, 2. Let K D be a compact subset with connected complement, and let fs) be a continuous function on K which is analytic in the interior of K. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup ζs + iτ, α) fs) < ε s K > 0. 0

11 First Theorem D by dierent methods has been obtained in [8] and [], see also [40]. We see that the approximated function fs), dierently from Theorem A, is not necessarily non-vanishing on K, and this is conditioned by non-existence of the Euler product over primes for the function ζs, α) in the case of Theorem D. We have that ζs, ) = ζs) and ζ s, ) = 2 s )ζs), 2 ) therefore, the functions ζs, ) and ζ s, 2 are also universal, however, the approximated function fs) must be non-vanishing on K. The case of algebraic irrational parameter α remains an open problem. Now we state a joint universality theorem for Hurwitz zeta-functions. Let, for 0 < α j, j =,..., r, Lα,..., α r ) = logm + α j ) : m N 0, j =,..., r. Theorem E. Suppose that the set Lα,..., α r ) is linearly independent over the eld of rational numbers Q. For j =,..., r, let K j D be a compact subset with connected complement, and let f j s) be a continuous function on K j which is analytic in the interior of K j. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup sup ζs + iτ, α j ) f j s) < ε s K j > 0. j r A proof of Theorem E is given in [23]. For algebraically independent over Q numbers α,..., α r α,..., α r are not roots of any polynomial px,..., x r ) 0 with rational coecients), Theorem E by a dierent method has been obtained in [36]. A generalization of the Hurwitz zeta-function is the periodic Hurwitz zeta- function introduced in [2]. Let a = a m : m N 0 be a periodic sequence of complex numbers with minimal period k N, and 0 < α. Then the periodic Hurwitz zeta-function ζs, α; a) is dened, for σ >, by ζs, α; a) = m=0 a m m + α) s. The periodicity of the sequence a implies, for σ >, the equality ζs, α; a) = k k s a l ζ s, l + α ). k l=0 This shows that the function ζs, α; a) also admits a meromorphic continuation with a simple pole at s = with residue a def = k a l. k In the case a = 0, the function ζs, α; a) is entire. l=0

12 In [2] and [3], the universality of the function ζs, α; a) with transcendental parameter α was investigated, and the following statement was proved. Theorem F. Suppose that the number α is transcendental. Let K D be a compact subset with connected complement, and let fs) be a continuous function on K which is analytic in the interior of K. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup ζs + iτ, α; a) fs) < ε s K > 0. Thus, Theorem F is an analogue of Theorem D in the case of transcendental α. The joint universality for periodic Hurwitz zeta-functions was began to study in [2]. For j =..., r, let a j = a mj : m N be a periodic sequence of complex numbers with minimal period k j N, α j, 0 < α j, be a xed parameter, and ζs, α j ; a j ) denote the corresponding periodic Hurwitz zeta-function. Denote by k the least common multiple of the periods k,..., k r, and dene the matrix a a 2... a r a A = 2 a a 2r a k a k2... a kr In [2], it was proved that if k j = k, α j = α for j =,..., r, α is transcendental, and ranka) = r, then the functions ζs, α, a ),..., ζs, α; a r ) are jointly universal. In [22], the requirement that k j = k for j =,..., r was removed. Finally, in [4] the following joint universality theorem was proved. Theorem G. Suppose that the numbers α,..., α r are algebraically independent over Q, and that ranka) = r. For j =,..., r, let K j and f j s) be the same as in Theorem E. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup sup ζs + iτ, α j ; a j ) f j s) < ε s K j > 0. j r It turned out that the rank hypothesis in Theorem G can be removed, and a joint universality theorem for periodic Hurwitz zeta-functions, without using the matrix A, forms Chapter of the thesis. Let Lα,..., α r ) be the same set as in Theorem E. We give a shortered statement of Theorem.. Theorem.. Suppose that the set Lα,..., α r ), is linearly independent over Q. For j =,..., r, let K j and f j s) be the same as in Theorem E. Then the assertion of Theorem G is true. The joint universality for periodic Hurwitz zeta-functions has a more general form when a collection of periodic sequences is attached to each parameter α j. First such an extension of the joint universality has been proposed in [3] for Lerch zeta-functions. The above idea for periodic Hurwitz zeta-functions has been applied in [24]. Let l j, j =,..., r, be positive integers. For every l =,..., l j, let 2

13 a jl = a mjl : m N 0 be a periodic sequence of complex numbers with minimal period k jl N. Suppose that, for j =,..., r, α j is a xed parameter, 0 < α j, and, for σ >, ζs, α j ; a jl ) = m=0 a mjl m + α j ) s. Denote by k the least common multiple of the periods k,..., k l,..., k r,..., k rlr, and dene the matrix a a 2... a l a a 2l2... a r a r2... a rlr a B = 2 a a 2l a a 22l2... a 2r a 2r2... a 2rlr a k a k2... a kl a k22... a k2l2... a kr a kr2... a krlr Moreover, let κ = r l j. j= Then in [24], the following result has been obtained. Theorem H. Suppose that the system Lα,..., α r ) is linearly independent over Q, and that rankb) = κ. For every j =,..., r, and l =,..., l j, let K jl be a compact subset of the strip D with connected complement, and let f jl s) be a continuous function on K jl which is analytic in the interior of K jl. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup sup j r l l j sup ζs + iτ, α j ; a jl ) f jl s) < ε s K jl > 0. In Chapter 2 of the thesis, the rank condition in Theorem H is made weaker. Let k j be the least common multiple of the periods k j, k j2,..., k jlj, j =,..., r. Dene a j a j2... a jlj a B j = 2j a 2j2... a 2jlj, j =,..., r a kjj a kjj2... a kjjl j Then the main result of Chapter 2 is the following theorem. Theorem 2.. Suppose that the set Lα,..., α r ) is linearly independent over Q, and that rankb j ) = l j, j =,..., r. Let K jl and f jl be the same as is Theorem H. Then the assertion of Theorem H is true. In Theorem 2., dierently from Theorem H, we use the information related only to α j, j =,..., r. All above joint universality theorems for zeta or L-functions are of the same type. Theorem C is an example of the joint universality for functions having the Euler product over primes, while all joint 3

14 theorems for periodic Hurwitz zeta-functions form a group of results for zeta-functions having no the Euler product. The paper of H. Mishou [35] is the rst work on the joint universality for zeta-functions of dierent types: having and having no the Euler product. We call this universality a mixed joint universality. In [35], a joint universality theorem for the Riemann and Hurwitz zeta-functions has been proved. Theorem I. Suppose that the number α is transcendental. Let K D, K 2 D be compact subsets with connected complements, f s) be a continuous non-vanishing function on K which is analytic in the interior of K, and let f 2 s) be a continuous function on K 2 which is analytic in the interior of K 2. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup ζs + iτ) f s) < ε, s K > 0. sup ζs + iτ, α) f 2 s) < ε s K 2 A generalization of Theorem I has been given in [6]. Let b = b m : m N be a periodic sequence of complex numbers with minimal period l N. Then the periodic zeta-function ζs; b) is dened, for σ >, by ζs; b) = m= b m m s. In view of periodicity of the sequence b, it follows that, for σ >, ζs; b) = l s l j= b j ζ s, j ), l and this gives meromorphic continuation for ζs; b) to the whole complex plane with possible pole at s = with residue If b = 0, then the function ζs; b) is entire. b def = l l b j. j= We recall that the sequence b is multiplicative if b =, and b mn = b m b n for all colprimes m, n N. The universality of the function ζs; b) with multiplicative sequence b has been obtained in [32]. In this case, the theorem is similar to Theorem A. In [6], the joint universality for the functions ζs; b) and ζs, α; a) has been obtained. Theorem J. Suppose that the sequence b is multiplicative such that, for every prime p, b p l c <, l= p l 2 4

15 and that the number α is transcendental. Let K, K 2, f s) and f 2 s) be the same as in Theorem I. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup ζs + iτ; b) f s) < ε, s K > 0. sup ζs + iτ, α; a) f 2 s) < ε s K 2 A multidimensional version of Theorem J is presented in [25]. In this case, the joint universality is obtained for the collection of zeta-functions ζs; b ),..., ζs; b r ) and ζs, α; a ),..., ζs, α r2 ; a r2 ). Theorem K [25]. Suppose that, for j =,..., r, the sequence b j is multiplicative such that, for every prime p, b p j l c j <, l= p l 2 and that the numbers α,..., α 2 are algebraically independent over Q. For j =,..., r, let K j D be a compact subset with connected complement, and let f j s) be a continuous non-vanishing function on K j which is analytic in the interior of K j. For j =,..., r 2, let K j D be a compact subset with connected complement, and let f j s) be a continuous function on K j which is analytic in the interior of K j. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup sup ζs + iτ; b j ) f j s) < ε, j r s K j sup sup ζs + iτ, α j ; a j ) f 2 s) < ε > 0. j r 2 s K j In Chapter 3 of the thesis, we generalize Theorem 2. adding to the functions ζs, α ; a ),..., ζs, α ; a l ),..., ζs, α r ; a r ),..., ζs, α r ; a rlr ) the Riemann zeta-function ζs). following statement. Thus, we have the Theorem 3.. Suppose that the numbers α,..., α r are algebraically independent over Q, rankb j ) = l j, j =,..., r, and that all hypotheses on the sets K jl and functions f jl s) of Theorem 2. hold. Moreover, let K D be a compact subset with connected complement, and let fs) be a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup ζs + iτ) fs) < ε, s K sup sup sup ζs + iτ, α j ; a jl ) f jl s) < ε > 0. s K jl j r l l j Chapter 4 of the thesis is devoted to a an analogue of Theorem 3. with the function ζs, F ) in place of the function ζs). Note that, in this case, we have a more complicated situation because 5

16 the functions ζs, F ) and ζs, α j ; a jl ) are universal in dierent strips D κ and D, respectively. We remind that here ζs, F ) denotes the zeta-function attached to a normalized Hecke eigen cusp form F of weight κ. Theorem 4.. Suppose that F is a normalized Hecke eigen cusp form of weight κ for the full modular group, the numbers α,..., α r are algebraically independent over Q, and that rankb j ) = l j, j =,..., r. Let K D κ be a compact subset with connected complement, fs) be a continuous non-vanishing function on K which is analytic in the interior of K, and that all hypothesis on the sets K jl and functions f jl of Theorem 2. hold. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup ζs + iτ, F ) fs) < ε, s K sup sup sup ζs + iτ, α j ; a jl ) f jl s) < ε > 0. s K jl j r l l j History of universality in analysis can be found in a very informative paper [9], see also [4], [20], [33]. The proofs of all joint universality theorems of the thesis are based on the probabilistic approach involving limit theorems for weakly convergent probability measures on the space of analytic functions, and on explicitly given supports of the limit measures. 6

17 Approbation The results of the thesis were presented at the Conferences of Lithuanian Mathematical Society ), at 0 th International Vilnius Conference on Probability Theory and Mathematical Statistics Vilnius, Lithuania, 28 th June -2 nd July, 200), at the 6 th International Conference Mathematical Modeling and Analysis Sigulda, Latvia, May 25-28, 20), at the 8 th International Algebraic Conference in Ukraine Lugansk, Ukraine, July 5-2, 20), at the International Conference 27 th Journées Arithmétiques Vilnius, Lithuania, 27 th June - st July, 20), as well as at the doctorant conferences of Institute of Mathematics and Informatics, and at the seminars of Number theory of Vilnius University and the seminar of the Faculty of Mathematics and Informatics of iauliai University. 7

18 Principal publications The main results of the thesis are published in the following papers:. A. Laurin ikas, S. Skerstonaite, A joint universality theorem for periodic Hurvitz zeta-functions. II, Lith. Math. J. Vol. 49, No 3, ). 2. A. Laurin ikas, S. Skerstonaite, Joint universality for periodic Hurwitz zeta-functions. II, in: New Directions in Value-Distribution Theory of Zeta and L-Functions, Würzburg Conference, 6-0 October 2008), R. Steuding and J. Steuding Eds.), Shaker Verlag, Aachen, ). 3. S. Ra kauskiene, D. iau i unas, Joint universality of some zeta functions. I, Lietuvos Matematikos Rinkinys. T. 5, ). 4. J. Genys, R. Macaitiene, S. Ra kauskiene, D. iau i unas,a mixed joint universality theorem for zeta-functions. Mathematical Modelling and Analysis, Vol. 5, No 4, ). 5. S. Ra kauskiene, D. iau i unas, A mixed joint universality theorem for zeta-functions. II, Submited). 8

19 Acknowledgment I would like to express my deep gratitude to my supervisor Professor Antanas Laurin ikas for his support and attention during the doctoral studies. Moreover, I would like to thank the members of the Numerical Analysis Department of the Institute of Mathematics and Informatics, of the Faculty of Mathematics and Informatics of iauliai University and of the Department of Probability Theory and Number Theory of Vilnius University for support and useful suggestions. 9

20 Chapter Joint universality for periodic Hurwitz zeta-functions In this chapter, we prove a joint universality theorem for periodic Hurwitz zeta-functions ζs, α ; a ),..., ζs, α r ; a r ). Here, for j =,..., r, α j, 0 < α j, is a xed parameter, a j = a mj : m N 0 is a periodic sequence of complex numbers with minimal period k N, and for σ >, If ζs, α j ; a j ) = m=0 a mj m + α j ) s. a j = k a lj 0, k j l=0 then the function ζs, α j ; a j ) is entire, while if a j 0, the function ζs, α j ; a j ) has a unique simple pole at s = with residue a j... Statement of the main theorem We recall that Lα,..., α r ) = logm + α j ) : m N 0, j =,..., r. For brevity, denote the elements of the set Lα,..., α r ) by e m,j = logm+α j ). The set Lα,..., α r ) is linearly independent over the eld of rational numbers Q if, for every nite collection e m,j,..., e mn,j l, m,..., m n N 0, j,..., j l,..., r, the equality q m,j e m,j + + q mn,j l e mn,j l = 0 20

21 with rationals q m,j,..., q mn,j l holds only in the case q m,j = = q mn,j l = 0. Obviously, in place of rationals q m,j,..., q mn,j l we may use rational integers. We remind that D = s C : 2 < σ <. Theorem.. Suppose that the set Lα,..., α r ) is linearly independent over Q. For j =,..., r, let K j D be a compact subset with connected complement, and let f j s) be a continuous function on K j which is analytic in the interior of K j. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup sup ζs + iτ, α j ; a j ) f j s) < ε > 0. s K j j r We note that Theorem. removes a certain rank condition on the coecients a mj which was used in [4]. A joint limit theorem on the weak convergence of probability measures in the space of analytic functions for periodic Hurwitz zeta-functions is the main ingredient in the proof of Theorem...2. Joint limit theorem We denote by HD) the space of analytic functions on D equipped with the topology of uniform convergence on compacta. In this topology, a sequence g n s) : n N HD) converges to the function gs) HD) if, for every compact subset K D, Dene lim sup g n s) gs) = 0. n s K H r D) = HD) HD). r Let, as usual, BS) denote the class of Borel sets of a space S. Moreover, let Ω = γ m, m=0 where γ m = s C : s = for all m N 0. Since the unit circle γ is a compact, by the Tikhonov theorem, see, for example, [37], the innite-dimensional torus Ω with the product topology and point wise multiplication is a compact topological Abelian group. Let Ω r = Ω Ω r, where Ω j = Ω for j =,..., r. Then Ω r also, by the Tikhonov theorem, is a compact topological Abelian group. Therefore [39], on Ω r, BΩ r )), the probability Haar measure m r H can be dened, and 2

22 we obtain the probability space Ω r, BΩ r ), m r H ). We remind that the measure mr H is invariant with respect to shifts by points from Ω r, i.e., m r HA) = m r HωA) = m r HAω) for every A BΩ r ) and all ω Ω. It is important to note that the Haar measure m r H is the product of the Haar measures m jh on the coordinate spaces Ω j, BΩ j )), j =,..., r. Denote by ω j m) the projection of an element ω j Ω j to the coordinate space γ m, m N 0, j =,..., r. Let ω = ω,..., ω r ) Ω r, where ω j Ω j, j =,..., r, and let, for brevity, α = α,..., α r ), a = a,..., a r ). On the probability space Ω r, BΩ r ), m r H ), dene the Hr D)-valued random element ζs, α, ω; a) by ζs, α, ω; a) = ζs, α, ω ; a ),..., ζs, α r, ω r ; a r ) ), where ζs, α j, ω j ; a j ) = m=0 a mj ω j m), j =,..., r. m + α j ) s We note that the latter series converges uniformly on compact subsets K D for almost all ω j Ω j, thus it denes an HD)-value random element, j =,..., r. Denote by P ζ the distribution of the random element ζs, α, ω; a), i.e., P ζ A) = m r H ω Ω r : ζs, α, ω; a) A ), A BH r D)). Let, for A BH r D)), P T A) = τ T meas [0, T ] : ζs + iτ, α; a) A, where ζs, α; a) = ζs, α ; a ),..., ζs, α r ; a r )). This section of the chapter is devoted to the following probabilistic limit theorem. Theorem.2. Suppose that the set Lα,..., α r ) is linearly independent over Q. Then P T converges weakly to the measure P ζ as T. We divide the proof of Theorem.2 into severe lemmas. The rst of them is a limit theorem on the torus Ω r. Let, for A BΩ r ), Q T A) = τ T meas [0, T ] : m + α ) iτ : m N 0 ),..., m + α r ) iτ : m N 0 )) A. 22

23 Lemma.3. Suppose that the set Lα,..., α r ) is linearly independent over Q. Then Q T converges weakly to the Haar measure m r H as T. A proof of Lemma.3 is based on the Fourier transform method on compact topological group, and is given in [24]. Now let σ > 2 be a xed number, and let, for m, n N 0, ) σ m + αj υ n m, α j ) = exp, j =,..., r. n + α j Dene and ζ n s, α j ; a j ) = ζ n s, α j, ω j ; a j ) = m=0 m=0 a mj υ n m, α j ) m + α j ) s, j =,..., r, a mj ω j m)υ n m, α j ) m + α j ) s, j =,..., r, It was proved in [2] that the latter series are absolutely convergent for σ > 2. The next important step in the proof of Theorem.2 are limit theorems in the space H r D) for the vectors and Let, for A BH r D)), and, for xed ω 0 Ω r, ζ n s, α; a) = ζ n s, α ; a ),..., ζ n s, α r ; a r )) ζ n s, α, ω; a) = ζ n s, α, ω ; a ),..., ζ n s, α r, ω r ; a r )). P T,n A) = τ T meas [0, T ] : ζ n s + iτ, α; a) A, Q T,n A) = τ T meas [0, T ] : ζ n s + iτ, α, ω 0 ; a) A. Lemma.4. Suppose that the set Lα,..., α r ) is linearly independent over Q. Then P T,n and Q T,n both converge weakly to the same probability measure P n on H r D), BH r D))) as T. Before the proof of Lemma.4, we remind the well-known fact from the theory of weak convergence of probability measures. Let S, BS )) and S 2, BS 2 )) be two measurable spaces, and h : S S 2 be a BS ), BS 2 ))-measurable function, i.e., for every A BS 2 ), h A BS ). Then every probability measure P on S, BS )) induces the unique probability measure P h on S 2, BS 2 )) dened by P h A) = P h A), A BS 2 ). 23

24 The following simple lemma which proof can be found in [3], Section 5, often is very useful. Lemma.5. Suppose that P n, n N, and P be probability measures on S, BS )), h : S S 2 be a continuous function, and let P n converges weakly to P as n. Then P n h also converges weakly to P h as n. Proof of Lemma.4. Since the series for ζ n s, α j ; a j ) and ζ n s, α j ω j ; a j ), j =,..., r, converges absolutely for σ > 2, the functions h n : Ω r H r D) and g n : Ω r H r D) given by h n ω) = ζ n s, α, ω; a) and g n ω) = ζ n s, α, ωω 0 ; a) are continuous. Moreover, we have that P T,n = Q T h n Q T,n = Q T gn. Therefore, from Lemmas.3 and.5 we obtain that P T,n and Q T,n converge weakly to m r H h n and m r H g n respectively, as T. Moreover, the invariance of the Haar measure m r H with respect to shifts by points from Ω r shows that m r Hg n = m r Hf n f 0 )) = m r Hf 0 f n where f 0 : Ω r Ω r is given by fω) = ωω 0, ω Ω r. ) = m r Hf 0 )fn = m r Hf n, In order to pass from ζ n s, α; a) to ζs, α; a), we need an approximation of ζs, α; a) and ζs, α, ω; a) by ζ n s, α; a) and ζ n s, α, ω; a), respectively. For this, we will use a metric on H r D) which induces its topology of uniform convergence on compacta. First, we dene such a metric on HD). For g, g 2 HD), we set ρg, g 2 ) = l= sup g s) g 2 s) 2 l s K l + sup g s) g 2 s), s K l where K l : l N is a sequence of compact subsets of D such that D = K l, l= K l K l+ for all l N, and if K D is a compact subset, then K K l for some l. The existence of such a sequence is given in [5]. Clearly, the metric ρ induces on HD) the topology of uniform convergence on compacta. Now, for g = g,..., g r ), g 2 = g 2,..., g 2r ) H r D), putting we obtain a desired metric on H r D). ρg, g 2 ) = max j r ρg j, g 2j ), and Lemma.6 The equality holds. lim lim sup n T T T 0 ρζs + iτ, α; a), ζ n s + iτ, α; a))dτ = 0 The proof of the lemma does not depend on arithmetical nature of the numbers α,..., α r, and is given in [3]. 24

25 Lemma.7 Suppose that the set Lα,..., α r ) is linearly independent over Q. Then, for almost all ω Ω r, we have the equality lim lim sup n T T T 0 ρζs + iτ, α, ω; a), ζ n s + iτ, α, ω; a))dτ = 0. Proof. Let a τ = m + α) iτ : m N 0, τ R, 0 < α, and dene ϕ τ : Ω Ω by ϕ τ ω) = a τ ω, ω Ω. Then ϕ τ : τ R is a one-parameter group of measurable measure - preserving transformations on Ω. A set A BΩ) is invariant with respect to the group ϕ τ : τ R if, for every τ R, the sets A and A τ = ϕ τ A) coincide up to a set of m H -measure zero, where m H is the probability Haar measure on Ω, BΩ)). The invariant sets form a σ-eld which is a σ-subeld of BΩ). A one-parameter groups ϕ τ : τ R is ergodic if its σ-eld of invariant sets consists only of sets of m H -measure zero or one. If the set Lα) = logm + α) : m N 0 is linearly independent over Q, then it is proved in [24] that the group ϕ τ : τ R is ergodic. Since the set Lα,..., α r ) is linearly independent over Q, so is each set Lα j ), j =,..., r. Combining this with the classical Birkho-Khintchine ergodic theorem, it is proved in [24] that, for every compact subset K D, lim lim sup n T T T 0 sup ζs + iτ, α j, ω j ; a j ) ζ n s + iτ, α j, ω j ; a j ) dτ = 0 s K for almost all ω j Ω j, j =,..., r. This and the denition of the metric ρ imply the equality lim lim sup n T T T 0 ρ ζs + iτ, α j, ω j ; a j ), ζ n s + iτ, α j, ω j ; j ) ) dτ = 0 for almost all ω j Ω j, j =,..., r, which, together with the denition of the metric ρ, yields the assertion of the lemma. For the proof of Theorem.2, we need one more lemma on a common limit measure. Let, for A BH r D)), P T A) = τ T meas [0, T ] : ζs + iτ, α, ω; a) A. Lemma.8. Suppose that the set Lα,..., α r ) is linearly independent over Q. Then P T P T, both converge weakly to the same probability measure P on H r D), BH r D))) as T. and The proof of the lemma is based on the Prokhorov theory of weak convergence of probability measures, therefore, rst we will remind some fact of that theory. Let P be a family of probability measures on S, BS)). The family P is called relatively compact if every sequence P n P contains a weakly convergent subsequence, and the family P is tight if, for every ε > 0, there exists a compact subset K S such that P K) > ε for all P P. tightness. The Prokhorov theorems connect the notions of the relative compactness and 25

26 Lemma.9. If the family of probability measures is tight, then it is relatively compact. Lemma.0. Suppose that the space S is complete and separable. If the family P is relatively compact, then it is tight. We also need one lemma from the theory of weak convergence of probability measures. Denote by D the convergence in distribution. Lemma.. Suppose that the space S, d) is separable, and Y n, X kn, k N, n N are S-valued random elements dened on the probability space Ω, B Ω), P). Let X kn D Xn as k, X n D X as n and, for every ε > 0, Then Y n D X as n. lim n lim sup P dx kn, Y n ) ε ) = 0. k The proofs of Lemmas.9-. can be found in [3]. Proof of Lemma.8. We take a random variable θ dened on a certain probability space Ω, B Ω), P) and uniformly distributed on [0, ]. X T,n = X T,n s) = X T,n, s),..., X T,n,r s)) = X T,n s, α; a) by Then we have, by Lemma.4, that On Ω, B Ω), P), dene the H r D)-valued random element X T,n s, α; a) = ζ n s + iθt, α; a). X T,n D T X n,.) where X n = X n s) = X n, s),..., X n,r s)), is an H r D)-valued random element having the distribution P n, and P n is the limit measure in Lemma.4. We will prove that the family of probability measures P n : n N 0 is tight. We have noted above that the series for ζ n s, α j ; a j ), j =,..., r, converges absolutely for σ > 2. Therefore, using the properties of the mean square of absolutely convergent Dirichlet series, we have that, for σ > 2, T lim ζn σ + it, α j ; a j ) 2 dt = T T 0 m=0 a mj 2 υ 2 nm, α j ) m + α j ) 2σ m=0 a mj 2 m + α j ) 2σ.2) for all n N 0 and j =,..., r. Let K l be a compact subset from the denition of the metric ρ. Then the Cauchy integral formula a standard application of the contour integration and.2), for all n N 0 and j =,..., r lead to the inequality T lim sup ζn s + iτ, α j ; a j ) dt Cl T T 0 s K l m=0 with some C l > 0 and σ l > 2. Now let ε > 0 be an arbitrary number, and a mj 2 R jl = m + α j ) 2σ l m=0 26 ) 2. a mj 2 m + α j ) 2σ l ) 2.3)

27 Then, taking M jl = C l R jl 2 l+r ε, we nd from.3) that ) lim sup P j : sup X T,n,j s) > M jl T s K l r M j= l T sup lim sup n N 0 T T 0 r j= ) lim sup P sup X T,n,j s) > M jl T s K l sup ζn s + iτ, α j ; a j ) dt s K l r j= C l R jl M jl < ε 2 l. This and.) show that, for all n N 0, ) P j : sup X n,j s) > M jl ε s K l 2 l..4) Dene a set H r ε = g,..., g r ) H r D) : sup g j s) M jl, j =,..., r, l N. s K l Then the set H r ε is uniformly bounded, thus, is a compact subset in the space H r D). Moreover, in view of.4), PX n s) H r ε ) ε l= 2 l = ε for all n N 0. Hence, by the denition of X n s), we nd that P n H r ε ) ε for all n N 0. This means that the family of probability measures P n : n N 0 is tight. Then, by Lemma.9, we have that the family P n : n N 0 is relatively compact. Therefore, there exists a subsequence P nk P n such that P n converges weakly to some probability measure P on H r D), BH r D))) as k, so the relation X nk D k P.5) holds. On Ω, B Ω, P), dene one more H r D)-valued random element X T = X T s, α; a) by Then, for every ε > 0, Lemma.6 implies that lim n X T s, α; a) = ζs + iθt, α; a). lim sup P ρx T s, α; a), X T,n s, α; a)) ε ) T = lim lim sup n T lim lim sup n T T meas τ [0, T ] : ρζs + iτ, α; a), ζ n s + iτ, α; a)) ε T ρζs + iτ, α; a), ζ n s + iτ, α; a)dτ = 0. T ε This, and relations.) and.5) together with Lemma. lead to 0 X T D P,.6) T 27

28 and this is equivalent to the weak convergence of P T to the measure P as T. Moreover, relation.6) shows that the probability measure P does not depend on the subsequence P nk. Hence, taking into account the relative compactness of the family P n : n N 0, we have that every subsequence of that family converges weakly to P, thus X n D P..7) n It remains to show that P T also converges weakly to the same measure P as T. For this, we dene the H r D)-valued random elements and X T,n s, α, ω; a) = ζ n s + iθt, α, ω; a) X T s, α, ω; a) = ζs + iθt, α, ω; a). Then, using.7) and Lemma.7, and repeating the above arguments for the random elements X T,n s, α, ω; a) and X T s, α, ω; a), we obtain the weak convergence of P T to the measure P as T. For the proof of Theorem.2, we recall an equivalent of the weak convergence of probability measures in terms of continuity sets. Let P be a probability measure on S, BS)), A BS), and let A denote the boundary of the set A. If P A) = 0, then the set A is called a continuity set of the measure P. Lemma.2. Let P and P n, n N, be probability measures on S, BS)). Then P n, as n, converges weakly to P if and only if, for every continuity set A of the measure P, lim P na) = P A). n Proof of the lemma is given in [8], Theorem 2.. We also need the classical Birkho-Khintchin ergodic theorem. Denote by Eξ the expectation of the random element ξ. Lemma.3. Suppose that Xt, ω) is an ergodic process, E Xt, ω) <, with sample paths integrable over every nite interval in the Riemann sense. Then, for almost all ω, lim T T T 0 Xt, ω)dt = EX0, ω). Proof of the lemma can be found, for example, in [6]. We state one more lemma from ergodicity theory. For τ R, dene a τ = m + α ) iτ : m N 0 ),..., m + α r ) iτ : m N 0 ), 28

29 and let Φ τ : τ R be the family of transformations on the torus Ω r given by Φ τ ω) = a τ ω, ω Ω r. Then Φ τ : τ R is a one-parameter group of measurable measure-preserving transformations on Ω r. The ergodicity of Φ τ : τ R is dened in the same way as that of the group ϕ τ : τ R used in the proof of Lemma.7. Lemma.4. The group Φ τ : τ R is ergodic. Proof of the lemma is given in [4], Lemma 3. Proof of Theorem.2. In view of Lemma.8, it is sucient to show that the limit measure P in that lemma coincides with P ζ. We x a continuity set A of the measure P in Lemma.8. Then, by Lemmas.8 and.2, we have that lim T T meas τ [0, T ] : ζs + iτ, α, ω; a) A = P A)..8) Let ξ be a random variable on the probability space Ω r, BΩ r ), m r H ) given by if ζs, α, ω; a) A, ξ = ξω) = 0 if ζs, α, ω; a) / A. In view of Lemma.4, we have that the random process ξφ τ ω)) is ergodic. Lemma.3, we obtain that, for almost all ω Ω r, Therefore, by lim T T T 0 ξφ τ ω))dτ = Eξ..9) On the other hand, the denition of ξ shows that Eξ = ξdm r H = m r Hω Ω r : ζs, α, ω; a) A), Ω r that is 0 Eξ = P ζ A)..0) Since, by the denitions of ξ and Φ τ, T ξφ τ ω))dτ = τ T T meas [0, T ] : ζs + iτ, α, ω; a) A, we see from relations.9) and.0) that, for almost all ω Ω r, lim T T meas τ [0, T ] : ζs + iτ, α, ω; a) A = P ζ A). This, together with.8), shows that P A) = P ζ A) for all continuity sets A of the measure P. However, all continuity sets constitute a determining class [3]. Thus, the measures P and P ζ coincide for all A BH r D)), and the theorem is proved. 29

30 .3. Support of the limit measure Denote by S Pζ the support of the measure P ζ. Since the space H r D) is separable, S Pζ is a minimal closed set of the space H r D) such that P ζ S Pζ ) =. The support S Pζ consists of all points g H r D) such that P ζ G) > 0 for every open neighbourhood G of g. Theorem.5. Suppose that the set Lα,..., α r ) is linearly independent over Q. support of the measure P ζ is the whole of H r D). Then the Proof. Let, for A j HD), j =,..., r, A = A A r. Since the space H r D) is separable, the σ-eld BH r D)) coincides with that generated by sets A [3]. Moreover, the Haar measure m r H is the product of the Haar measures m H,..., m rh. Therefore, P ζ A) = m r Hω Ω r : ζs, α, ω; a) A) = m r Hω Ω r : ζs, α, ω; a) A A r ) = m r Hω Ω r : ζs, α, ω ; a ) A,..., ζs, α r, ω r ; a r ) A r ) = m H ω Ω : ζs, α, ω ; a ) A )... m rh ω r Ω : ζs, α r, ω r ; a r ) A r )..) Since the set Lα,..., α r ) is linearly independent over Q, so is each set Lα ),..., Lα r ). Therefore, we have from [3] that, for every j =,..., r, the support of m jh ω j Ω j : ζs, α j, ω j ; a j ) A) is the whole of HD). Thus, the latter remark and.) prove the theorem..4. Proof of Theorem. We start with the famous Mergelyan theorem on approximation of analytic functions by polynomials. Lemma.6. Let K C be a compact subset with connected complement, and let fs) be a continuous function on K which is analytic in the interior of K. Then, for every ε > 0, there exists a polynomial ps) such that sup fs) ps) < ε. s K Proof of the lemma is given in [34], see also [44]. 30

31 sets. We also remind an equivalent of the weak convergence of probability measures in terms of open Lemma.7. Let P and P n, n N, be probability measures on S, BS)). Then P n, as n, converges weakly to P if and only if, for every open set G of S, lim P ng) P G). n Let Proof of the lemma can be found in [3], Theorem 2.. Proof of Theorem.. In witue of Lemma.6, there exist polynomials p s),..., p r s) such that G = sup j r g,..., g r ) H r D) : sup sup f j s) p j s) < ε s K j 2..2) j r sup g j s) p j s) < ε s K j 2 Clearly, G is an open set. Moreover, in view of Theorem.5, p s),..., p r s)) S Pζ. Therefore, by properties of a support mentioned in the beginning of Section.3, the inequality P ζ G) > 0 holds. By Theorem. and Lemma.7, we have that lim inf T P T G) P ζ G). Thus, we deduce from the denitions of the set G and P T that lim inf T T meas τ [0, T ] : sup sup ζs + iτ, α j ; a j ) p j s) < ε > 0..3) s K j 2 However, inequalities.2) and j r. imply sup j r sup ζs + iτ, α j ; a j ) p j s) < ε s K j 2 Therefore, sup j r τ [0, T ] : τ [0, T ] : sup ζs + iτ, α j ; a j ) f j s) < ε. s K j sup j r sup j r sup ζs + iτ, α j ; a j ) f j s) < ε s K j sup ζs + iτ, α j ; a j ) p j s) < ε s K j 2 This, together with inequality.3), yields that lim inf T T meas τ [0, T ] : sup sup ζs + iτ, α j ; a j ) f j s) < ε j r s K j lim inf T T meas τ [0, T ] : sup sup ζs + iτ, α j ; a j ) p j s) < ε > 0. s K j 2 The theorem is proved. j r. 3

32 Chapter 2 Extended joint universality theorem for periodic Hurwitz zeta-functions The aim of this chapter is an extension of Theorem. for a wider collection of periodic Hurwitz zetafunctions. Let l j, j =,..., r, be positive integers, and, for l =,..., l j, let a jl = a mjl : m N 0 be a periodic sequence of complex numbers with minimal period k jl N. Suppose that, for j =,..., r, α j is a xed parameter, 0 < α j, and that ζs, α j ; a jl ) is the corresponding periodic Hurwitz zeta-function. In this chapter, we consider the joint universality for the functions ζs, α ; a ),..., ζs, α ; a l ),..., ζs, α r ; a r ),..., ζs, α r ; a rlr ). 2.. Statement of an extended joint universality theorem Theorem. was obtained without any hypotheses on the coecients of the functions ζs, α ; a,..., ζs, α r ; a r ). However, in the case when a collection of periodic sequences corresponds each parameter α j, we need a certain rank condition. Let k j be the common multiple of the periods k j,..., k jlj, j =,..., r. Dene a j a j2... a jlj a B j = 2j a 2j2... a 2jlj, j =,..., r a kjj a kjj2... a kjjl j Then the main theorem of the chapter is of the form. 32

33 Theorem 2.. rankb j ) = l j, j =,..., r. Suppose that the set Lα,..., α r ) is linearly independent over Q, and that For every j =,..., r and l =,..., l j, let K jl be a compact subset of the strip D with connected complement, and let f jl s) be a continuous function on K jl which is analytic in the interior of K jl. Then, for every ε > 0, lim inf T T meas τ [0, T ] : sup sup sup ζs + iτ, α j ; a jl ) f jl s) < ε > 0. s K jl j r l l j 2.2. Extended joint limit theorem The proof of Theorem 2., as that of Theorem., is based on a probability joint limit theorems for the functions ζs, α ; a ),..., ζs, α ; a l ),..., ζs, α r ; a r ),..., ζs, α r ; a rlr ). For brevity, let a = a,..., a l,..., a r,..., a rlr ), and κ = r j= l j, H κ D) = HD) HD). We preserve κ the notation used in Chapter. On the probability space Ω r, BΩ r ), m r H ), dene the Hκ D)-valued random element ζ κ s, α, ω; a) by ζ κ s, α, ω; a) = ζs, α, ω ; a ),..., ζs, α r, ω r ; a r ),..., ζs, α r, ω r ; a rlr )), where ζs, α l, ω l ; a jl ) = m=0 a mjl ω j m) m + α j ) s, j =,..., l j. Denote by P ζ,κ the distribution of the random element ζs, α, ω; a), i.e., P ζ,κ A) = m r Hω Ω r : ζ κ s, α, ω; a) A), A BH κ D)). In this section, we consider the weak convergence, as T, for P T,κ A) = τ T meas [0, T ] : ζ κ s + iτ, α; a) A, A BH κ D)), where ζ κ s, α; a) = ζs, α ; a ),..., ζs, α ; a l ),..., ζs, α r ; a r ),..., ζs, α r ; a rlr )). Theorem 2.2. Suppose that the set Lα,..., α r ) is linearly independent over Q. converges weakly to the measure P ζ,κ as T. Then P T,κ We see that the statement of Theorem 2.2 does not contain the hypothesis on the rank of the matrices B j, therefore, the proof of Theorem 2.2 remains similar to that of Theorem.2. For this reason, we will present only the principal steps of the proof. 33

34 and We dene ζ n s, α j ; a jl ) = ζ n s, α j, ω j ; a jl ) = m=0 m=0 a mjl υ n m, α j ) m + α j ) s, j =,..., r, l =,..., l j, a mjl ω j m)υ n m, α j ) m + α j ) s, j =,..., r, l =,..., l j. Since the coecients a mjl are bounded, the latter series, as those for the functions ζ n s, α j ; a j ) and ζ n s, α j, ω j ; a j ) in Section.2, are absolutely convergent for σ > 2. We start with limit theorems in the space H κ D) for and ζ n,κ s, α; a) = ζ n s, α ; a ),..., ζ n s, α ; a l ),..., ζ n s, α r ; a r ),..., ζ n s, α r ; a rlr )) ζ n,κ s, α, ω; a) = ζ n s, α, ω ; a ),..., ζ n s, α, ω ; a l ),..., ζ n s, α r, ω r ; a r ),..., ζ n s, α r, ω r ; a rlr )). For A BH κ D)), dene P T,n,κ A) = τ T meas [0, T ] : ζ n,κ s + iτ, α; a) A, and, for any xed ω 0 Ω r, Q T,n,κ A) = τ T meas [0, T ] : ζ n,κ s + iτ, α, ω 0 ; a) A. Lemma 2.3. Suppose that the set Lα,..., α r ) is linearly independent over Q. Then P T,n,κ and Q T,n,κ both converge weakly to the same probability measure P n,κ on H κ D), BH κ D))) as T. Proof. The lemma uses Lemmas.3 and.5, and is obtained in the same way as Lemma.4. Let ρ be the same metric on HD) as in Section.2. For f = f,..., f l,..., f r,..., f rlr ), g = g,..., g l,..., g r,..., g rlr ) H κ D), dene ρ κ f, g) = max max ρf jl, g jl ). j r j l j Then ρ κ is a,metric on H κ D) which induces the topology of uniform convergence on compacta. Two next lemmas give an approximation in the mean for ζ κ s, α; a) by ζ n,κ s, α; a) as well as for ζ κ s, α, ω; a) by ζ n,κ s, α, ω; a). holds. Lemma 2.4. The equality lim lim sup n T T T 0 ρ κ ζ κ s + iτ, α; a), ζ n,κ s + iτ, α; a))dτ = 0 34

35 Proof of the lemma is given in [24], Lemma 2. Using the estimate [2] T 0 ζσ + it, α j ; a jl ) 2 dt = OT ), σ > 2, j =,..., r, l =,..., l j, rst it is proved that, for every compact subset K D, T lim lim sup sup ζs + iτ, α j ; a jl ) ζ n s + iτ, α j ; a jl ) dτ = 0, n T T 0 s K j =,..., r, l =,..., l j. From this and the denition of the metric ρ κ, the lemma follows. Lemma 2.5. Suppose that the set Lα,..., α r ) is linearly independent over Q. Then, for almost all ω Ω r, the equality holds. lim lim sup n T T T 0 ρ κ ζ κ s + iτ, α, ω; a), ζ n,κ s + iτ, α, ω; a))dτ = 0 Proof of the lemma is given in [24], Lemma 5. We note that the linear independence of the set Lα,..., α r ) is not exhausted fully, the linear independence of the sets Lα ),..., Lα r ) is sucient. The ergodicity of group ϕ τ : τ R dened in the proof of Lemma.7 leads, for almost all ω j Ω j, to the estimate T 0 ζσ + it, α j, ω j ; a jl ) 2 dt = OT ), σ > 2, j =,..., r, l =,..., l j. From this estimate, by a standard contour integration method it is derived that, for every compact subset K D, lim lim sup n T T T 0 sup ζs + iτ, α j, ω j ; a jl ) ζ n s + iτ, α j, ω j ; a jl ) dτ = 0 s K for almost all ω j Ω j, j =,..., r, l =,..., l j. Combining the latter relation with the denition of the metric ρ κ gives the assertion of the lemma. Dene one more probability measure P T,κ A) = τ T meas [0, T ] : ζ κ s + iτ, α, ω; a) A, A BH κ D)). Lemma 2.6. Suppose that the set Lα,..., α r ) is linearly independent over Q. Then P T,κ and P T,κ both converge weakly to the same probability measure P κ on H κ D), BH κ D))) as T. Proof. Let θ be the same random variable as in the proof of Lemma.8. On Ω, B Ω, P), dene the H κ D)-valued random element X T,n,κ = X T,n,κ s) = X T,n,κ s, α; a) = X T,n,, s),..., X T,n,,l s),..., X T,n,r, s),..., X T,n,r,lr s)) by Then, in view of Lemma 2.3, we have that X T,n,κ s, α; a) = ζ n,κ s + iθt, α; a). X T,n,κ D T X n,κ, 2.) 35