- Email: [email protected]
Universality of Various Zeta-Functions
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 43 (2013) 129–135 www.elsevier.com/locate/endm
Universality of Various Zeta-Functions Roma Kaˇcinskait˙e 1,2 Department of Mathematics ˇ Siauliai University ˇ Siauliai, Lithuania
Abstract This is a survey of results on joint universality in Voronin’s sense of various zetafunctions, when in the collection of these functions some of them have the Euler product and the others have not. Keywords: analytic function, joint approximation, periodic sequence, universality, zeta-function.
In the number theory, the investigation of universality of zeta-functions is interesting and important subject of the studies. In 1975, S. M. Voronin proved that every analytic non-vanishing function on compact subsets can be approximated by the shifts of the Riemann zetafunction ζ(s), s = σ + it, which, for σ > 1, is defined by ∞ 1 1 −1 ζ(s) = = 1− s ms p m=1 p 1
Partially supported by the European Commission within the 7th Framework Programme FP/2011–2014 project INTEGER (INstitutional Transformation for Effecting Gender Equality in Research), Grant Agreement No. 266638. 2 Email:[email protected] 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.07.022
130
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
(here p denotes a prime number). Now this property we call as universality. Theorem 1.1 ([12]) Let 0 < r < 14 , and let f (s) be any non-vanishing continuous function on the disc |s| ≤ r which is analytic in the interior of this disc. Then, for every ε > 0, there exists a number τ = τ (ε) ∈ R such that 3 max ζ s + + iτ − f (s) < ε. |s|≤r 4 We can state it in more general form. Let D = {s ∈ C :
1 2
< σ < 1}.
Theorem 1.2 ([6]) Let K be a compact subset of the strip D with connected complement. Let f (s) be a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every ε > 0, 1 lim inf meas τ ∈ [0, T ] : sup |ζ(s + iτ ) − f (s)| < ε > 0. T →∞ T s∈K Last inequality shows that the set of translations of the Riemann zetafunction which approximate a given analytic function f (s) has positive lower density. Many authors generalized the Voronin theorem for other zeta- and Lfunctions. We can mentioned B. Bagchi, S. M. Gonek, A. Reich, A. Lauˇ ci¯ rinˇcikas, K. Matsumoto, H. Nagoshi, J. Steuding, D. Siauˇ unas, the author and other (for history and results, see, for example, [3], [7], [10]). There exists a conjecture of Linnik-Ibragimov that all functions in some half-plane defined by Dirichlet series, analytically continuable to the left of absolute convergence half-plane and satisfying some natural growth conditions are universal in Voronin sense. The first result on joint approximation of a given collection of analytic functions by a collection of shifts of zeta-functions belongs to S. M. Voronin [11]. He investigated the collection of Dirichlet L-functions L(s, χ). We recall that the function L(s, χ) attached to a character χ mod d, d ∈ N, for σ > 1, is given −1 ∞ χ(m) χ(p) L(s, χ) = 1− s = . ms p m=1 p Theorem 1.3 ([11]) Let χ1 , . . . , χn be pairwise non-equivalent Dirichlet characters, and L(s, χ1 ), . . . , L(s, χn ) are the corresponding Dirichlet L-functions. For j = 1, . . . , n, let Kj denote a compact subset of the strip D with connected complement, and fj (s) be a continuous non-vanishing function on Kj and
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
131
analytic in the interior of Kj . Then, for every ε > 0,
1 lim inf meas τ ∈ [0, T ] : sup sup |L(s + iτ, χj ) − fj (s)| < ε > 0. T →∞ T 1≤j≤n s∈Kj More complicated situation we have in the two-dimensional or multidimensional case when in the same collection part of the zeta-functions have Euler product but the other do not have. In 2007, H. Mishou proved the joint universality theorem for the Riemann zeta-function ζ(s) and Hurwitz zetafunction ζ(s, α) with a transcendental parameter α [8]. It is well-known that the function ζ(s, α), 0 < α ≤ 1, for σ > 1, is defined by ζ(s, α) =
∞
1 . s (m + α) m=0
It has an analytic continuation to the whole complex plane except a simple pole at s = 1 with residue 1. If α = 1, then the Hurwitz zeta-function ζ(s, 1) reduces to the Riemann zeta-function ζ(s). Then the following statement holds. Theorem 1.4 ([8]) Suppose that α is a transcendental number such that 0 < α < 1. Let K1 and K2 be compact subsets of the strip 12 < σ < 1 with connected complements. Assume that functions fj (s) are continuous on Kj and analytic in the interior of Kj for each j = 1, 2. In addition, we suppose that f1 (s) does not vanish on K1 . Then, for all positive ε, lim inf T →∞
1 meas(τ ∈ [0, T ] : max |ζ(s + iτ ) − f1 (s)| < ε, s∈K1 T max |ζ(s + iτ, α) − f2 (s)| < ε) > 0. s∈K2
The joint approximation of a given collection of analytic functions by a collection of shifts of periodic zeta-function and periodic Hurwitz zeta-function is obtained by A. Laurinˇcikas and the author in [4]. Let a = {am : m ∈ N} be a periodic sequence of complex numbers with the least period k ∈ N. The periodic zeta-function ζ(s; a), for σ > 1, is defined by the series ∞ am ζ(s; a) = , s m m=1 and by analytic continuation elsewhere. From the periodicity of sequence a
132
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
follows that, for σ > 1, k m 1 . am ζ s, ζ(s; a) = s k m=1 k
Last equality gives an analytic continuation to the whole complex plane for the kfunction ζ(s; a), except, maybe for the point s = 1 with residue a = 1 m=1 am . If a = 0, then ζ(s; a) is an entire function. k If the sequence a is completely multiplicative, then the periodic zetafunction ζ(s; a) coincides with the Dirichlet L-function L(s, χ). The periodic Hurwitz zeta-function ζ(s, α; b) with a fixed parameter α, 0 < α ≤ 1, is defined, for σ > 1, by ζ(s, α; b) =
∞
bm , (m + α)s m=0
where b = {bm : m ∈ N ∪ {0}} is a periodic sequence of complex numbers bm with a minimal period l ∈ N. From the periodicity of b, for σ > 1, we have l−1 m+α 1 . bm ζ s, ζ(s, α; b) = s l m=0 l This gives an analytic continuation of the function ζ(s, α; b) to the l−1whole 1 complex plane, except, for a simple pole at s = 1 with residue b = l m=0 bm . If b = 0, then periodic Hurwitz zeta-function is an entire function. Theorem 1.5 ([4]) Suppose that α is a transcendental number. Let K1 and K2 be a compact subsets of the strip D with connected complements, f1 (s) be a continuous non-vanishing function on K1 which is analytic in the interior of K1 , and let f2 (s) be a continuous function on K2 which is analytic in the interior of K2 . Then, for every ε > 0, 1 lim inf meas τ ∈ [0, T ] : sup |ζ(s + iτ ; a) − f1 (s)| < ε, T →∞ T s∈K1 sup |ζ(s + iτ, α; b) − f2 (s)| < ε > 0. s∈K2
Multidimensional version of Theorem 1.5 is obtained by A. Laurinˇcikas in [5].
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
133
Theorem 1.6 ([5]) Suppose that the sequences a1 , ..., ar1 are multiplicative, and, for all prime p, holds the inequality ∞ |ajpg | j=1
pg/2
< 1,
j = 1, ..., r1 .
Let α1 , ..., αr2 be algebraically independent over Q. Suppose that K1 , ..., Kr1 ˆ 1 , ..., K ˆ r2 are compact subsets of the strip D, their complements are conand K nected. Suppose that f1 (s), ..., fr1 (s) are continuous non-vanishing functions in K1 , ..., Kr1 and analytic in interior K1 , ..., Kr1 , and fˆ1 (s), ..., fˆr2 (s) are conˆ 1 , ..., K ˆ r2 and analytic in interior K ˆ 1 , ..., K ˆ r2 , respectively. Then, tinuous in K for every ε > 0, 1 lim inf meas τ ∈ [0, T ] : sup sup |ζ(s + iτ ; aj ) − fj (s)| < ε, T →∞ T 1≤j≤r1 s∈Kj sup sup |ζ(s + iτ, αj ; bj ) − fˆj (s)| < ε > 0. 1≤j≤r2 s∈K ˆj
All universality theorems stated above are of continuous type: we deal with mathematical objects given by integrals. In discrete case, the trigonometric and other sums appear. Therefore, discrete universality theorems are more complicated: there the shifts are taken from certain arithmetic progression with the step h > 0. First result in discrete approximation of analytic functions belongs to A. Reich [9]. In 2011, the author obtains joint discrete universality of Dirichlet L-function L(s, χ) and periodic Hurwitz zeta-function ζ(s, α; b) with transcendental parameter α [1]. Theorem 1.7 ([1]) Suppose that α, K1 , K2 , f1 (s) and f2 (s) are the same as in Theorem 1.5. Let h > 0 be a fixed number such that exp{ 2π } is rational. h Then, for every ε > 0, 1 # 0 < r ≤ N : sup |L(s + irh, χ) − f1 (s)| < ε, lim inf N →∞ N + 1 s∈K1 sup |ζ(s + irh, α; b) − f2 (s)| < ε > 0. s∈K2
It is possible to generalize Theorem 1.7 and obtain the joint universality of collection of Dirichlet L-functions and periodic Hurwitz zeta-functions with transcendental parameters. Theorem 1.8 ([2]) Let h > be a fixed number such that exp{ 2π } is ratioh nal. Suppose that χ1 , . . . , χr1 are pairwise non-equivalent Dirichlet characters,
134
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
and L(s, χ1 ), . . . , L(s, χr1 ) are the corresponding Dirichlet L-functions. Let α1 , . . . , αr2 be algebraically independent over Q. Suppose that K1 , . . . , Kr1 , ˆ 1, . . . , K ˆ r2 , fˆ1 (s), . . . , fˆr2 (s) satisfy the hypothesis of Theof1 (s), . . . , fr1 (s), K rem 1.6. Then, for every ε > 0, 1 lim inf # 0 < r ≤ N : sup sup |L(s + irh, χj ) − fj (s)| < ε, N →∞ N + 1 1≤j≤r1 s∈Kj
sup sup |ζ(s + irh, αj ; bj ) − fˆj (s)| < ε > 0 1≤j≤r2 s∈K ˆj
References [1] Kaˇcinskait˙e, R., Joint Discrete Universality of Periodic Zeta-Functions, Int. Transf. Spec. Funct. 22(8) (2011), 593–601. [2] Kaˇcinskait˙e, R., Joint Discrete Universality of Periodic Zeta-Functions. II, submitted. [3] Kaˇcinskait˙e, R., Limit Theorems for Zeta-Functions – with Application in ˇ Universality, Siauliai Math. Semin. 7(15) (2012), 19–40. [4] Kaˇcinskait˙e, R., and Laurinˇcikas, A., The Joint Distribution of Periodic ZetaFunctions, Stud. Sci. Math. Hung. 48(2) (2011), 257–279. [5] Laurinˇcikas, A., Joint Universality of Zeta-Functions with Periodic Coefficients, Izv. Math. 74(3) (2010), 515–539 = Izv. Ross. Akad. Nauk, Ser. Mat. 74(3), (2010) 79–102 (in Russian). [6] Laurinˇcikas, A., “Limit Theorems for the Riemann Zeta-Function”, Kluwer, Dordrecht, 1996. [7] Matsumoto, K., Probabilistic Value-Distrubtion Theory of Zeta-Functions, Sugaku Expositions 17(1) (2004), 51–71. [8] Mishou, H., The Joint Value-Distribution of the Riemann Zeta-Function and Hurwitz Zeta-Functions, Liet. Matem. Rink. 47(1) (2007), 62–807 = Lith. Math. J. 47(1)(2007), 32–47. [9] Reich, A., Zur Universalit¨ at und Hypertranszendenz der Dedekindschen Zetafunktion, Abh. Braunschweig. Wiss. Ges. 33 (1982), 197–203. [10] Steuding, J., “Value-Distribution of L-Functions”, Springer-Verlag, Berlin, Heidelberg, New York, 2007.
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
135
[11] Voronin, S. M., On the Functional Independence of Dirichlet L-Functions, Acta Arith. 27 (1975), 493–503 (in Russian). [12] Voronin, S. M., Theorem on the “Universality” of the Riemann Zeta-Function, Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), 475–486 (in Russian) = Math. USSR Izv. 9 (1975), 443–453.
Electronic Notes in Discrete Mathematics 43 (2013) 129–135 www.elsevier.com/locate/endm
Universality of Various Zeta-Functions Roma Kaˇcinskait˙e 1,2 Department of Mathematics ˇ Siauliai University ˇ Siauliai, Lithuania
Abstract This is a survey of results on joint universality in Voronin’s sense of various zetafunctions, when in the collection of these functions some of them have the Euler product and the others have not. Keywords: analytic function, joint approximation, periodic sequence, universality, zeta-function.
In the number theory, the investigation of universality of zeta-functions is interesting and important subject of the studies. In 1975, S. M. Voronin proved that every analytic non-vanishing function on compact subsets can be approximated by the shifts of the Riemann zetafunction ζ(s), s = σ + it, which, for σ > 1, is defined by ∞ 1 1 −1 ζ(s) = = 1− s ms p m=1 p 1
Partially supported by the European Commission within the 7th Framework Programme FP/2011–2014 project INTEGER (INstitutional Transformation for Effecting Gender Equality in Research), Grant Agreement No. 266638. 2 Email:[email protected] 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.07.022
130
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
(here p denotes a prime number). Now this property we call as universality. Theorem 1.1 ([12]) Let 0 < r < 14 , and let f (s) be any non-vanishing continuous function on the disc |s| ≤ r which is analytic in the interior of this disc. Then, for every ε > 0, there exists a number τ = τ (ε) ∈ R such that 3 max ζ s + + iτ − f (s) < ε. |s|≤r 4 We can state it in more general form. Let D = {s ∈ C :
1 2
< σ < 1}.
Theorem 1.2 ([6]) Let K be a compact subset of the strip D with connected complement. Let f (s) be a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every ε > 0, 1 lim inf meas τ ∈ [0, T ] : sup |ζ(s + iτ ) − f (s)| < ε > 0. T →∞ T s∈K Last inequality shows that the set of translations of the Riemann zetafunction which approximate a given analytic function f (s) has positive lower density. Many authors generalized the Voronin theorem for other zeta- and Lfunctions. We can mentioned B. Bagchi, S. M. Gonek, A. Reich, A. Lauˇ ci¯ rinˇcikas, K. Matsumoto, H. Nagoshi, J. Steuding, D. Siauˇ unas, the author and other (for history and results, see, for example, [3], [7], [10]). There exists a conjecture of Linnik-Ibragimov that all functions in some half-plane defined by Dirichlet series, analytically continuable to the left of absolute convergence half-plane and satisfying some natural growth conditions are universal in Voronin sense. The first result on joint approximation of a given collection of analytic functions by a collection of shifts of zeta-functions belongs to S. M. Voronin [11]. He investigated the collection of Dirichlet L-functions L(s, χ). We recall that the function L(s, χ) attached to a character χ mod d, d ∈ N, for σ > 1, is given −1 ∞ χ(m) χ(p) L(s, χ) = 1− s = . ms p m=1 p Theorem 1.3 ([11]) Let χ1 , . . . , χn be pairwise non-equivalent Dirichlet characters, and L(s, χ1 ), . . . , L(s, χn ) are the corresponding Dirichlet L-functions. For j = 1, . . . , n, let Kj denote a compact subset of the strip D with connected complement, and fj (s) be a continuous non-vanishing function on Kj and
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
131
analytic in the interior of Kj . Then, for every ε > 0,
1 lim inf meas τ ∈ [0, T ] : sup sup |L(s + iτ, χj ) − fj (s)| < ε > 0. T →∞ T 1≤j≤n s∈Kj More complicated situation we have in the two-dimensional or multidimensional case when in the same collection part of the zeta-functions have Euler product but the other do not have. In 2007, H. Mishou proved the joint universality theorem for the Riemann zeta-function ζ(s) and Hurwitz zetafunction ζ(s, α) with a transcendental parameter α [8]. It is well-known that the function ζ(s, α), 0 < α ≤ 1, for σ > 1, is defined by ζ(s, α) =
∞
1 . s (m + α) m=0
It has an analytic continuation to the whole complex plane except a simple pole at s = 1 with residue 1. If α = 1, then the Hurwitz zeta-function ζ(s, 1) reduces to the Riemann zeta-function ζ(s). Then the following statement holds. Theorem 1.4 ([8]) Suppose that α is a transcendental number such that 0 < α < 1. Let K1 and K2 be compact subsets of the strip 12 < σ < 1 with connected complements. Assume that functions fj (s) are continuous on Kj and analytic in the interior of Kj for each j = 1, 2. In addition, we suppose that f1 (s) does not vanish on K1 . Then, for all positive ε, lim inf T →∞
1 meas(τ ∈ [0, T ] : max |ζ(s + iτ ) − f1 (s)| < ε, s∈K1 T max |ζ(s + iτ, α) − f2 (s)| < ε) > 0. s∈K2
The joint approximation of a given collection of analytic functions by a collection of shifts of periodic zeta-function and periodic Hurwitz zeta-function is obtained by A. Laurinˇcikas and the author in [4]. Let a = {am : m ∈ N} be a periodic sequence of complex numbers with the least period k ∈ N. The periodic zeta-function ζ(s; a), for σ > 1, is defined by the series ∞ am ζ(s; a) = , s m m=1 and by analytic continuation elsewhere. From the periodicity of sequence a
132
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
follows that, for σ > 1, k m 1 . am ζ s, ζ(s; a) = s k m=1 k
Last equality gives an analytic continuation to the whole complex plane for the kfunction ζ(s; a), except, maybe for the point s = 1 with residue a = 1 m=1 am . If a = 0, then ζ(s; a) is an entire function. k If the sequence a is completely multiplicative, then the periodic zetafunction ζ(s; a) coincides with the Dirichlet L-function L(s, χ). The periodic Hurwitz zeta-function ζ(s, α; b) with a fixed parameter α, 0 < α ≤ 1, is defined, for σ > 1, by ζ(s, α; b) =
∞
bm , (m + α)s m=0
where b = {bm : m ∈ N ∪ {0}} is a periodic sequence of complex numbers bm with a minimal period l ∈ N. From the periodicity of b, for σ > 1, we have l−1 m+α 1 . bm ζ s, ζ(s, α; b) = s l m=0 l This gives an analytic continuation of the function ζ(s, α; b) to the l−1whole 1 complex plane, except, for a simple pole at s = 1 with residue b = l m=0 bm . If b = 0, then periodic Hurwitz zeta-function is an entire function. Theorem 1.5 ([4]) Suppose that α is a transcendental number. Let K1 and K2 be a compact subsets of the strip D with connected complements, f1 (s) be a continuous non-vanishing function on K1 which is analytic in the interior of K1 , and let f2 (s) be a continuous function on K2 which is analytic in the interior of K2 . Then, for every ε > 0, 1 lim inf meas τ ∈ [0, T ] : sup |ζ(s + iτ ; a) − f1 (s)| < ε, T →∞ T s∈K1 sup |ζ(s + iτ, α; b) − f2 (s)| < ε > 0. s∈K2
Multidimensional version of Theorem 1.5 is obtained by A. Laurinˇcikas in [5].
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
133
Theorem 1.6 ([5]) Suppose that the sequences a1 , ..., ar1 are multiplicative, and, for all prime p, holds the inequality ∞ |ajpg | j=1
pg/2
< 1,
j = 1, ..., r1 .
Let α1 , ..., αr2 be algebraically independent over Q. Suppose that K1 , ..., Kr1 ˆ 1 , ..., K ˆ r2 are compact subsets of the strip D, their complements are conand K nected. Suppose that f1 (s), ..., fr1 (s) are continuous non-vanishing functions in K1 , ..., Kr1 and analytic in interior K1 , ..., Kr1 , and fˆ1 (s), ..., fˆr2 (s) are conˆ 1 , ..., K ˆ r2 and analytic in interior K ˆ 1 , ..., K ˆ r2 , respectively. Then, tinuous in K for every ε > 0, 1 lim inf meas τ ∈ [0, T ] : sup sup |ζ(s + iτ ; aj ) − fj (s)| < ε, T →∞ T 1≤j≤r1 s∈Kj sup sup |ζ(s + iτ, αj ; bj ) − fˆj (s)| < ε > 0. 1≤j≤r2 s∈K ˆj
All universality theorems stated above are of continuous type: we deal with mathematical objects given by integrals. In discrete case, the trigonometric and other sums appear. Therefore, discrete universality theorems are more complicated: there the shifts are taken from certain arithmetic progression with the step h > 0. First result in discrete approximation of analytic functions belongs to A. Reich [9]. In 2011, the author obtains joint discrete universality of Dirichlet L-function L(s, χ) and periodic Hurwitz zeta-function ζ(s, α; b) with transcendental parameter α [1]. Theorem 1.7 ([1]) Suppose that α, K1 , K2 , f1 (s) and f2 (s) are the same as in Theorem 1.5. Let h > 0 be a fixed number such that exp{ 2π } is rational. h Then, for every ε > 0, 1 # 0 < r ≤ N : sup |L(s + irh, χ) − f1 (s)| < ε, lim inf N →∞ N + 1 s∈K1 sup |ζ(s + irh, α; b) − f2 (s)| < ε > 0. s∈K2
It is possible to generalize Theorem 1.7 and obtain the joint universality of collection of Dirichlet L-functions and periodic Hurwitz zeta-functions with transcendental parameters. Theorem 1.8 ([2]) Let h > be a fixed number such that exp{ 2π } is ratioh nal. Suppose that χ1 , . . . , χr1 are pairwise non-equivalent Dirichlet characters,
134
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
and L(s, χ1 ), . . . , L(s, χr1 ) are the corresponding Dirichlet L-functions. Let α1 , . . . , αr2 be algebraically independent over Q. Suppose that K1 , . . . , Kr1 , ˆ 1, . . . , K ˆ r2 , fˆ1 (s), . . . , fˆr2 (s) satisfy the hypothesis of Theof1 (s), . . . , fr1 (s), K rem 1.6. Then, for every ε > 0, 1 lim inf # 0 < r ≤ N : sup sup |L(s + irh, χj ) − fj (s)| < ε, N →∞ N + 1 1≤j≤r1 s∈Kj
sup sup |ζ(s + irh, αj ; bj ) − fˆj (s)| < ε > 0 1≤j≤r2 s∈K ˆj
References [1] Kaˇcinskait˙e, R., Joint Discrete Universality of Periodic Zeta-Functions, Int. Transf. Spec. Funct. 22(8) (2011), 593–601. [2] Kaˇcinskait˙e, R., Joint Discrete Universality of Periodic Zeta-Functions. II, submitted. [3] Kaˇcinskait˙e, R., Limit Theorems for Zeta-Functions – with Application in ˇ Universality, Siauliai Math. Semin. 7(15) (2012), 19–40. [4] Kaˇcinskait˙e, R., and Laurinˇcikas, A., The Joint Distribution of Periodic ZetaFunctions, Stud. Sci. Math. Hung. 48(2) (2011), 257–279. [5] Laurinˇcikas, A., Joint Universality of Zeta-Functions with Periodic Coefficients, Izv. Math. 74(3) (2010), 515–539 = Izv. Ross. Akad. Nauk, Ser. Mat. 74(3), (2010) 79–102 (in Russian). [6] Laurinˇcikas, A., “Limit Theorems for the Riemann Zeta-Function”, Kluwer, Dordrecht, 1996. [7] Matsumoto, K., Probabilistic Value-Distrubtion Theory of Zeta-Functions, Sugaku Expositions 17(1) (2004), 51–71. [8] Mishou, H., The Joint Value-Distribution of the Riemann Zeta-Function and Hurwitz Zeta-Functions, Liet. Matem. Rink. 47(1) (2007), 62–807 = Lith. Math. J. 47(1)(2007), 32–47. [9] Reich, A., Zur Universalit¨ at und Hypertranszendenz der Dedekindschen Zetafunktion, Abh. Braunschweig. Wiss. Ges. 33 (1982), 197–203. [10] Steuding, J., “Value-Distribution of L-Functions”, Springer-Verlag, Berlin, Heidelberg, New York, 2007.
R. Kaˇcinskait˙e / Electronic Notes in Discrete Mathematics 43 (2013) 129–135
135
[11] Voronin, S. M., On the Functional Independence of Dirichlet L-Functions, Acta Arith. 27 (1975), 493–503 (in Russian). [12] Voronin, S. M., Theorem on the “Universality” of the Riemann Zeta-Function, Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), 475–486 (in Russian) = Math. USSR Izv. 9 (1975), 443–453.