Ramanujan J DOI 10.1007/s11139-014-9631-2

A discrete version of the Mishou theorem Eugenijus Buivydas · Antanas Laurinˇcikas

Received: 17 January 2014 / Accepted: 19 August 2014 © Springer Science+Business Media New York 2014

Abstract H. Mishou proved that the Riemann zeta-function and Hurwitz zetafunction with transcendental parameter are jointly universal, i.e., their shifts (continuous) approximate any pair of analytic functions. In the paper, a discrete version of the Mishou theorem is presented. In this case, the parameter of the Hurwitz zeta-function and the step of discrete shifts are connected by a certain independence relation. Keywords Algebraically independent numbers · Hurwitz zeta-function · Joint universality · Limit theorem · Riemann zeta-function · Universality Mathematics Subject Classification

11M06 · 11M35

1 Introduction Let s = σ +it be a complex variable. The Riemann zeta-function ζ (s) and the Hurwitz zeta-function ζ (s, α) with parameter α, 0 < α ≤ 1, are defined, for σ > 1, by

Partially supported by grant from the Research Council of Lithuania. E. Buivydas Department of Mathematics and Informatics, Šiauliai University, P. Višinskio 19, 77156 Šiauliai, Lithuania e-mail: [email protected] A. Laurinˇcikas (B) Department of Mathematics and Informatics, Vilnius University, Naugarduko st. 24, 03225 Vilnius, Lithuania e-mail: [email protected]

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 ∞   1 −1 1 1− s ζ (s) = = ms p p m=1

and ζ (s, α) =

∞  m=0

1 , (m + α)s

where the infinite product is taken over all prime numbers p. Moreover, the functions ζ (s) and ζ (s, α) have analytic continuations to the whole complex plane, except for simple poles at the point s = 1 with residue 1. It is well known that the functions ζ (s) and ζ (s, α) for some classes of the parameter α are universal in the sense that their shifts ζ (s + iτ ) and ζ (s + iτ, α), τ ∈ R, approximate any analytic function uniformly on compact sets of the right-hand side of the critical strip. Universality of the Riemann zeta-function was discovered by S. M. Voronin in 1975 [18]. We give the latest version of the Voronin theorem. Let D = {s ∈ C : 21 < σ < 1}. Denote by K the class of compact subsets of the strip D with connected complements, and by H0 (K ), K ∈ K, the class of continuous nonvanishing functions on K which are analytic in the interior of K . Then we have, see for example [7], [17], the following statement. Theorem 1 Suppose that K ∈ K and f (s) ∈ H0 (K ). Then, for every ε > 0, lim inf T →∞

  1 meas τ ∈ [0, T ] : sup |ζ (s + iτ ) − f (s)| < ε > 0. T s∈K

Here and in the sequel, measA denotes the Lebesgue measure of a measurable set A ⊂ R. From the definition of the function ζ (s, α), we have that ζ (s, 1) = ζ (s), and ζ (s, 21 ) = (2s − 1)ζ (s). The universality property of ζ (s, α) depends on the parameter α, and differs a bit from Theorem 1. Denote by H (K ), K ∈ K, the class of continuous functions on K which are analytic in the interior of K . Then we have Theorem 2 Suppose that the number α is transcendental or rational = 1, 21 . Let K ∈ K and f (s) ∈ H (K ). Then, for every ε > 0, lim inf T →∞

  1 meas τ ∈ [0, T ] : sup |ζ (s + iτ, α) − f (s)| < ε > 0. T s∈K

Theorem 2 for rational α = 1, 21 was proved independently by Voronin [19], Gonek [5] and Bagchi [1]. The case of transcendental α can be found in [9]. The function ζ (s, α) is also universal with α = 1 (Theorem 1) and α = 21 , however, in this case, the

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A discrete version of the Mishou theorem

approximated function belongs to the class H0 (K ). The case of algebraic irrational α is an open problem. In [12], H. Mishou obtained an interesting joint universality theorem on the approximation of a pair of analytic functions by shifts (ζ (s + iτ ), ζ (s + iτ, α)). Theorem 3 [12]. Suppose that the number α is transcendental. Let K 1 , K 2 ∈ K, and f 1 (s) ∈ H0 (K 1 ) and f 2 (s) ∈ H (K 2 ). Then, for every ε > 0,  1 lim inf meas τ ∈ [0, T ] : sup |ζ (s + iτ ) − f 1 (s)| < ε, T →∞ T s∈K 1 sup |ζ (s + iτ, α) − f 2 (s)| < ε > 0. s∈K 2

Theorems 1–3 show that the sets of shifts ζ (s + iτ ), ζ (s + iτ, α) and (ζ (s + iτ ), ζ (s + iτ, α)) approximating given analytic functions are infinite and even have a positive lower density. Theorems 1–3 are of the so-called continuous type. In shifts ζ (s + iτ ) and ζ (s + iτ, α) τ varies continuously in the interval [0, T ]. Also, the discrete universality is considered. In this case, analytic functions are approximated by shifts ζ (s + ikh) and ζ (s + ikh, α), k ∈ N0 = N ∪ {0}, and h > 0 is a fixed number. A discrete analogue of Theorem 1 is of the form. Theorem 4 Suppose that K ∈ K and f (s) ∈ H0 (K ). Then, for every ε > 0, lim inf N →∞

  1 # 0 ≤ k ≤ N : sup |ζ (s + ikh) − f (s)| < ε > 0. N +1 s∈K

The discrete universality of ζ (s) was proved by A. Reich in [15]. Theorem 4 under slightly different conditions on the set K was also obtained by B. Bagchi [1]. The discrete universality for the Hurwitz zeta-function is more complicated because in this case two parameters α and h occur, and a connection between them plays an important role. Theorem 5 Suppose that the number α is transcendental or rational = 1, 21 , K ∈ K and f (s) ∈ H (K ). In the case of rational α, let the number h > 0 be arbitrary, while

inthe case of transcendental α, let the number h > 0 be such that the number exp 2π h is rational. Then, for every ε > 0, lim inf N →∞

  1 # 0 ≤ m ≤ N : sup |ζ (s + imh, α) − f (s)| < ε > 0. N +1 s∈K

Theorem 5 for rational α was obtained in [1]. J. Sander and J. Steuding [16] gave a different proof. In the case of transcendental α, Theorem 5 is a particular case of a theorem from [10]. Universality of zeta-functions has a series theoretical and practical applications. For practical applications, the discrete universality is more convenient. For example, a

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discrete universality theorem was applied [3] for estimation of complicated integrals over analytic curves which are considered in quantum mechanics. This is a motivation together with continuous universality also to investigate the discrete universality of zeta-functions. The aim of this paper is a discrete version of Theorem 3. Denote by P the set of all prime numbers, an define the set 

L(P, α, h) = (log p : p ∈ P), (log(m + α) : m ∈ N0 ), 2π h Theorem 6 Suppose that the set L(P, α, h) is linearly independent over the field of rational numbers Q. Let K , K 1 ∈ K, and f (s) ∈ H0 (K ), f 1 (s) ∈ H (K ). Then, for every ε > 0,  1 lim inf # 0 ≤ k ≤ N : sup |ζ (s + ikh) − f (s)| < ε, N →∞ N + 1 s∈K sup |ζ (s + ikh, α) − f 1 (s)| < ε > 0. s∈K 1

We give some examples of the numbers α and h satisfying the hypothesis of Theorem 6. Suppose that the numbers α and exp{ 2π h } are algebraically independent over Q. Then the set L(P, α, h) is linearly independent over Q. Really, it is well known that the set {log p : p ∈ P} is linearly independent over Q. Since the numbers α and exp{ 2π h } are algebraically independent, they are transcendental. Therefore, if we have that k1 log p1 + · · · + km log pm + l1 log(m 1 + α) + · · · + lr log(m r + α) = 0, where not all k j ∈ Z and l j ∈ Z are zeros, we obtain that km (m 1 + α)l1 · · · (m r + α)lr = 1, p1k1 · · · pm

and this contradicts the transcendence of the number α. If k1 log p1 + · · · + km log pm + l1 log(m 1 + α) + · · · +lr log(m r + α) + l 2π h =0

(1)

with l ∈ Z \ {0} and at least one l j ∈ Z \ {0}, then

l km (m 1 + α)l1 · · · (m r + α)lr exp{ 2π p1k1 · · · pm h } = 1, and this contradicts the algebraic independence of the numbers α and exp{ 2π h }. If, in equality (1), l ∈ Z \ {0}, all l j = 0 and at last one k j ∈ Z \ {0}, then the equality

l km exp{ 2π p1k1 · · · pm h } =1

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A discrete version of the Mishou theorem

contradicts the transcendence of the number exp{ 2π h }. For example, by the Nesterenko theorem [14], the numbers π and eπ are algebraically independent over Q. Thus, the numbers α = π√−1 and h√ = 2 satisfy the hypothesis of Theorem 6. Similarly, since 3 3 the numbers 2 2 and 2 4 are algebraically independent over Q [4], we may take   √ −1 3 2π and h = √ . α= 2 2 3 4 log 2

2 A joint discrete limit theorem For the proof of Theorem 6, we will use a probabilistic approach based on limit theorems on weakly convergent probability measures in the space of analytic functions. Denote by γ the unit circle on the complex plane, and define two tori 1 =



∞ 

γ p , 2 =

p

γm ,

m=0

where γ p = γ for all primes p, and γm = γ for all non-negative integers m. The tori 1 and 2 with the product topology and pointwise multiplication are compact topological Abelian groups. Therefore, in virtue of the classical Tikhonov theorem, = 1 × 2 is also a compact topological Abelian group. Denote by B(X ) the Borel σ -field of the space X . Then, on ( , B( )), the probability Haar measure m H can be defined, and we obtain the probability space ( , B( ), m H ). We note that m H is the product of probability Haar measures m 1H and m 2H on ( 1 , B( 1 )) and ( 2 , B( 2 )), respectively. We start with a discrete limit theorem on the torus . Define   1 # 0 ≤ k ≤ N : ( p −ikh : p ∈ P), N +1   ((m + α)−ikh : m ∈ N0 ) ∈ A , A ∈ B( ).

Q N (A) =

Lemma 1 Suppose that the set L(P, α, h) is linearly independent over Q. Then Q N converges weakly to the Haar measure m H as N → ∞. Proof Denote by ω1 ( p) the projection of an element ω1 ∈ 1 to γ p , p ∈ P, and by ω2 (m) the projection of an element ω2 ∈ 2 to γm , m ∈ N0 . The dual group of is isomorphic to the group de f

G =





⊕ Zp ⊕

p∈P



 ⊕ Zm ,

m∈N0

where Z p = Z for all p ∈ P

and Zm = Z for all m ∈ N0 . An element (k, l) = (k p : p ∈ P), (lm : m ∈ N0 ) of G, where only a finite number of integers k p and lm

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are distinct from zero, acts on by the formula k



l

(ω1 , ω2 ) → (ω1 , ω2 ) =

k

ω1 p ( p)

p∈P



ω2lm (m).

m∈N0

Therefore, the Fourier transform g N (k, l) of the measure Q N is ⎛

 g N (k, l) =

⎝



k

ω1 p ( p)

p∈P

⎞



ω2lm (m)⎠ d Q N

m∈N0

N 1   −ikk p h  p (m + α)−iklm h N +1 k=0 p∈P m∈N0 ⎧ ⎛ ⎞⎫ N ⎨ ⎬   1  = exp −ikh ⎝ k p log p + lm log(m + α)⎠ , (2) ⎩ ⎭ N +1

=

p∈P

k=0

m∈N0

where, as above, only a finite number of integers k p and lm are distinct from zero. The linear independence of the set L(P, α, h) implies that of the set {(log p : p ∈ P), (log(m + α) : m ∈ N0 )}. Therefore, 

k p log p +

p∈P



lm log(m + α) = 0

m∈N0

if and only if k = 0 and l = 0. Moreover, ⎧ ⎨

⎛

exp −i h ⎝ ⎩





k p log p +

p∈P

m∈N0

⎞⎫ ⎬ lm log(m + α)⎠ = 1 ⎭

(3)

for (k, l) = (0, 0). Indeed, if the latter inequality is not true, then

exp

⎧ ⎨ ⎩

k p log p +

p∈P

 m∈N0

⎫ ⎬



2πr lm log(m + α) = exp ⎭ h



with some non-zero integer r . However, this contradicts the linear independence of the set L(P, α, h). From (2) and (3), we derive that g N (k, l) = ⎧ 1 ⎪ ⎪   ⎪ ⎪ ⎨ 1−exp −i(N +1)h  ⎪ ⎪ ⎪ ⎪ ⎩ (N +1)

123





1−exp −i h



p∈P

 p∈P

k p log p+

 

m∈N0

if (k, l) = (0, 0),



if (k, l) = (0, 0).

lm log(m+α)

m∈N0

k p log p+



lm log(m+α)

A discrete version of the Mishou theorem

Therefore,  lim g N (k, l) =

N →∞

1 if (k, l) = (0, 0), 0 if (k, l) = (0, 0).

Hence, by a continuity theorem for probability measures on compact groups, see for example Theorem 1.4.2 of [6], we have that the measure Q N converges weakly to m H as N → ∞.  Let θ >

1 2

be a fixed number, for m, n ∈ N,     m θ , vn (m) = exp − n

and, for m ∈ N0 , n ∈ N,    m+α θ vn (m, α) = exp − . n+α Define the series ζn (s) =

∞ ∞   vn (m) vn (m, α) , ζ (s, α) = n s m (m + α)s

m=1

m=0

and ζn (s, ω1 ) =

∞  ω1 (m)vn (m) , ms

ζn (s, α, ω2 ) =

m=1

∞  ω2 (m)vn (m, α) , (m + α)s

m=0

where ω1 (m) =



ω1l ( p),

m ∈ N.

pl |m pl+1 m

Then it is well known, see for example [7] and [9], that the above series are absolutely convergent for σ > 21 . Let, for brevity, ω = (ω1 , ω2 ), ζ n (s, α) = (ζn (s), ζn (s, α)) and ζ n (s, α, ω) = (ζn (s, ω1 ), ζn (s, α, ω2 )) .

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Denote by H (D) the space of analytic functions on D endowed with the topology of uniform convergence on compacta, and, on the space (H 2 (D), B(H 2 (D))), define two probability measures   1 # 0 ≤ k ≤ N : ζ n (s + ikh, α) ∈ A N +1

de f

PN ,n (A) = and, for ω ∈ ,

  1 # 0 ≤ k ≤ N : ζ n (s + ikh, α, ω) ∈ A . N +1

de f PˆN ,n (A) =

Lemma that

2 Suppose

the set L(P, α, h) is linearly independent over Q. Then, on 2 H (D) , B H 2 (D) , there exists a probability measure Pn such that the measures PN ,n and PˆN ,n both converge weakly to Pn as N → ∞. Proof Let the function h n : → H 2 (D) be given by the formula  h n (ω) =

∞ ∞  ω1 (m)vn (m)  ω2 (m)vn (m, α) , . ms (m + α)s

m=1

m=0

Since the series for ζn (s, ω1 ) and ζn (s, α, ω2 ) converge absolutely for σ > 21 , we have that the function h n is continuous. Moreover,   h n ( p −ikh : p ∈ P), ((m + α)−ikh : m ∈ N0 ) = ζ n (s + ikh, α). Therefore, the continuity of h n , Lemma 1 and Theorem 5.1 of [2] imply that the measure PN ,n converges weakly to m H h −1 n as N → ∞. Since the measure m H is invariant with respect to shifts by points from , repeating of similar arguments shows that the measure PN ,n also converges weakly to m H h −1 n as N → ∞.  Let ζ (s, α) = (ζ (s), ζ (s, α)) and ζ (s, α, ω) = (ζ (s, ω1 ), ζ (s, α, ω2 )) , where ζ (s, ω1 ) =

∞ ∞   ω1 (m) ω2 (m) and ζ (s, α, ω ) = . 2 ms (m + α)s

m=1

123

m=0

A discrete version of the Mishou theorem

The latter series converge uniformly on compact subsets of the strip D for almost all ω1 ∈ 1 [7] and for almost all ω2 ∈ 2 [9], respectively. Thus, ζ (s, α, ω) is defined correctly. We also note that ζ (s, α, ω) is the H 2 (D)-valued random element defined on the probability space ( , B( ), m H ). We will approximate ζ (s, α) by ζ n (s, α) and ζ (s, α, ω) by ζ n (s, α, ω) in the mean. Let ρ be a metric on H (D) inducing the topology of uniform convergence on compacta, for the definition, see [7]. Let, for g 1 = (g11 , g12 ) ∈ H 2 (D) and g 2 = (g21 , g22 ) ∈ H 2 (D), ρ(g 1 , g 2 ) = max (ρ(g11 , g21 ), ρ(g12 , g22 )) . Then ρ is a metric on H 2 (D) inducing its topology. Lemma 3 The equality

lim lim sup

n→∞ N →∞

N  1   ρ ζ (s + ikh, α), ζ n (s + ikh, α) = 0 N +1 k=0

holds. Proof From the definition of the metric ρ, it follows that it is sufficient to prove that 1  lim lim sup ρ (ζ (s + ikh), ζn (s + ikh)) = 0 n→∞ N →∞ N + 1 N

(4)

k=0

and

1  ρ (ζ (s + ikh, α), ζn (s + ikh, α)) = 0. N +1 N

lim lim sup

n→∞ N →∞

(5)

k=0

The function ζn (s) has [7] the integral representation 1 ζn (s) = 2πi



θ+i∞

θ−i∞

ζ (s + z)ln (z)

1 dz , σ > , z 2

(6)

where ln (s) = θs

s θ

ns ,

and (s) is the Euler gamma-function. Let K be a compact subset of the strip D. Then an application of the residue theorem and (6) leads to the estimate N 1  sup |ζ (s + ikh) − ζn (s + ikh)| N +1 s∈K m=0

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E. Buivydas, A. Laurinˇcikas



∞

−∞

 |ln (σˆ + iτ )|

1  |ζ (σ + it + ikh + iτ )| dτ N +1 N

(7)

k=0

with σˆ < 0, 1 > σ > we find that

1 2

and bounded t. By the Gallagher lemma [13], Lemma 1.4,

1  |ζ (σ + it + ikh + iτ )| N +1 N

k=0



N 1  |ζ (σ + it + ikh + iτ )|2 N

1 2

1 + |τ |.

k=0

This and (7) show that 1  sup |ζ (s + ikh) − ζn (s + ikh)| = 0. N +1 s∈K N

lim lim sup

n→∞ N →∞

k=0

Hence, by the definition of the metric ρ, we obtain equality (4). Equality (5) is a particular case of Theorem 4.1 from [10], and of the definition of ρ.  Lemma 4 Suppose that the set L(P, α, h) is linearly independent over Q. Then, for almost all ω ∈ , the equality

lim lim sup

n→∞ N →∞

N  1   ρ ζ (s + ikh, α, ω), ζ n (s + ikh, α, ω) = 0 N +1 k=0

holds. Proof As in the case of Lemma 3, it suffices to show that, for almost all ω1 ∈ 1 , 1  ρ (ζ (s + ikh, ω1 ), ζn (s + ikh, ω1 )) = 0, N +1 N

lim lim sup

n→∞ N →∞

(8)

k=0

and, for almost all ω2 ∈ 2 , 1  ρ (ζ (s + ikh, α, ω2 ), ζn (s + ikh, α, ω2 )) = 0. N +1 N

lim lim sup

n→∞ N →∞

123

k=0

(9)

A discrete version of the Mishou theorem

In [7], it is proved that, for σ > 

T

1 2

and almost all ω1 ∈ 1 ,

|ζ (σ + it, ω1 )|2 dt = O(T ).

0

Therefore, using the Gallagher lemma, we obtain that, for 1 > σ > ω1 ∈ 1 ,

1 2

and almost all

1  |ζ (σ + it + ikh + iτ, ω1 )| 1 + |τ | N +1 N

k=0

for bounded t. Therefore, the proof of (8) runs in a similar way as that of (4). Also, it is known [9] that, for 1 > σ > 21 and almost all ω2 ∈ 2 , 

T

|ζ (σ + it, α, ω2 )|2 dt = O(T )

(10)

0

with transcendental α. However, the transcendence of α is used only for the linear independence over Q of the set L(α) = {log(m + α) : m ∈ N0 } . The linear independence of the set L(P, α, h) implies that of L(α). Therefore, under hypothesis of the lemma, estimate (10) is valid. An application of (10) and the Gallagher lemma leads, for 21 < σ < 1 and almost all ω2 ∈ 2 , to the estimate 1  |ζ (σ + it + ikh + iτ, ω2 )| 1 + |τ | N +1 N

k=0



for bounded t. The latter estimate implies relation (9). Now we will consider the weak convergence of the probability measures PN (A) =

  1 # 0 ≤ k ≤ N : ζ (s + ikh, α) ∈ A , N +1

  A ∈ B H 2 (D) ,

and PˆN (A) =

  1 # 0 ≤ k ≤ N : ζ (s + ikh, α, ω) ∈ A , N +1

  A ∈ B H 2 (D) .

Lemma 5 Suppose that the set L(P, α, h) is linearly independent over Q. Then, on (H 2 (D), B(H 2 (D))), there exists a probability measure P such that the measures PN and PˆN both converge weakly to P as N → ∞.

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E. Buivydas, A. Laurinˇcikas

Proof By Lemma 2, the measure P

N ,n converges weakly to Pn as N → ∞. Let X n = X n (s) = X n,1 (s), X n,2 (s) be the H 2 (D)-valued random element having the distribution Pn , and let θ N be a discrete random variable defined on a certain ˆ B( ), ˆ P) and having the distribution probability space ( , P (θ N = kh) =

1 , k = 0, 1, . . . , N . N +1

Consider the H 2 (D)-valued random element X N ,n = X N ,n (s) = (X N ,n,1 (s), X N ,n,2 (s)) defined by X N ,n (s) = ζ n (s + iθ N , α). D

Then, denoting by − → the convergence in distribution, we can rewrite the assertion of Lemma 2 in the form D X N ,n (s) −−−−→ X n (s). (11) N →∞

The absolute convergence for σ > 21 of the series for ζn (s) and ζn (s, α) together with the Gallagher lemma implies the boundedness of the discrete mean squares of these functions. From this and (11), we deduce by a standard way that the family of probability measures {Pn : n ∈ N} is tight. Therefore, by the Prokhorov theorem [2], 6.1, this family is relatively compact, and there exists a subsequence 

Theorem Pn k ⊂ {Pn } such that Pn k converges weakly to a certain probability measure P on (H 2 (D), B(H 2 (D))) as k → ∞. Obviously, this is equivalent to D

X n k −−−→ P. k→∞

(12)

ˆ B( ), ˆ P), define one more H 2 (D)-valued random eleOn the probability space ( , ment X N = X N (s) = ζ (s + iθ N , α) . Then, in virtue of Lemma 3, we find that, for every ε > 0,  

lim lim sup P ρ X N (s), X N ,n (s) ≥ ε = 0.

n→∞ N →∞

This, (11) and (12) show that all hypotheses of Theorem 4.2 from [2] are satisfied, therefore, D

X N −−−−→ P, N →∞

or the measure PN converges weakly to P as N → ∞. Moreover, the latter relation

 implies that the measure P is independent of the choice of the subsequence Pn k .

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A discrete version of the Mishou theorem

 Thus, in view of the relative compactness of the family Pn k , D

X n −−−→ P.

(13)

n→∞

It remains to prove that the measure PˆN also converges weakly to P as N → ∞. ˆ B( ), ˆ P), define two H 2 (D)-valued random For this, on the probability space ( , elements Xˆ N ,n = Xˆ N ,n (s) = ζ n (s + iθ N , α, ω) and Xˆ N = Xˆ N (s) = ζ (s + iθ N , α, ω). Then, repeating the above arguments for the random elements Xˆ N ,n and Xˆ N , and using Lemma 4 and relation (13), we find that the measure PˆN also converges weakly to P as N → ∞.  Now we are in position to prove the main limit theorem for ζ (s, α). Denote by Pζ the distribution of the random element ζ (s, α, ω), i. e., for A ∈ B(H 2 (D)),   Pζ (A) = m H ω ∈ : ζ (s, α, ω) ∈ A . Theorem 7 Suppose that the set L(P, α, h) is linearly independent over Q. Then the measure P in Lemma 5 coincides with Pζ . Proof We start with one transformation on the torus . Let ah,α =



   p −i h : p ∈ P , (m + α)−i h : m ∈ N0 .

Define a transformation f h,α (ω) of the torus by f h,α (ω) = ah,α ω, ω ∈ . Then f h,α is a measurable measure preserving transformation on ( , B( ), m H ). We recall that a set A ∈ B( ) is called invariant with respect to the transformation f h,α if the sets A and Ah,α = f h,α (A) differ one from another at most by the set of zero m H -measure. The transformation f h,α is called ergodic if its σ -field of invariant sets consists of sets having m H -measure equal to 0 or 1. We will prove that the transformation f h,α is ergodic. For this, we will apply once more inequality (3). In the proof of Lemma 1, we already have used that a character χ of is of the form χ (ω) =

 p∈P

k

ω1 p ( p)



ω2lm (m),

m∈N0

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E. Buivydas, A. Laurinˇcikas

where only a finite number of integers k p and lm are distinct from zero. First let χ be a non-trivial character. Then, by (3), ⎧ ⎛ ⎞⎫ ⎨ ⎬   χ (ah,α ) = exp −i h ⎝ k p log p + lm log(m + α)⎠ = 1, ⎩ ⎭ p∈P

(14)

m∈N0

for (k, l) = (0, 0). The further arguments, using (14), deal with the Fourier transform of the indicator function I A (ω), where A is an invariant set of transformation f h,α , and are analogical to those of Lemma 5.1 from [10]. The remainder part of the proof is standard. An equivalent of the weak convergence of probability measures in terms of continuity sets for the measure PˆN of Lemma 5 is combined with a discrete version of the classical Birkhoff-Khintchine theorem whose application is possible in virtue of ergodicity of the transformation f h,α . The details with different notation can be found in [10], Theorem 6.1.  3 Support of Pζ The space H 2 (D) is separable, thus, the support of Pζ is a minimal closed set Sζ ⊂ H 2 (D) such that Pζ (Sζ ) = 1. Moreover, the set Sζ consists of all g ∈ H 2 (D) such that, for every open neighbourhood G of g, the inequality Pζ (G) > 0 holds. Let S = {g ∈ H (D) : g(s) = 0 or g(s) ≡ 0} . Then we have the following statement. Theorem 8 Suppose that the set L(P, α, h) is linearly independent over Q. Then the support of Pζ is the set S × H (D). Proof Since the space H (D) is separable, we have [2] that B(H 2 (D)) = B(H (D)) × B(H (D)). Hence, it suffices to consider Pζ on the sets A = A1 × A2 ,

A1 , A2 ∈ H (D).

We have noted above that the Haar measure m H is the product of the Haar measures m 1H and m 2H . Therefore,   Pζ (A) = m H ω ∈ : ζ (s, α, ω) ∈ A1 × A2 = m 1H (ω1 ∈ 1 : ζ (s, ω1 ) ∈ A1 ) m 2H (ω2 ∈ 2 : ζ (s, α, ω2 ) ∈ A2 ) . (15)

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It is known [7] that the support of the random element ζ (s, ω1 ) is the set S. The linear independence of the set L(P, α, h) implies that of the set L(α) = {log(m + α) : m ∈ N0 }. Therefore [8], the support of the random element ζ (s, α, ω2 ) is the whole of H (D). Let Pζ be the distribution of ζ (s, ω1 ) and Pζ,α be the distribution of ζ (s, α, ω2 ), i.e., Pζ (A1 ) = m 1H (ω1 ∈ 1 : ζ (s, ω1 ) ∈ A1 ) and Pζ,α (A2 ) = m 2H (ω2 ∈ 2 : ζ (s, α, ω2 ) ∈ A2 ) . Then, in virtue of (15), Pζ (A) = Pζ (A1 )Pζ,α (A2 ).

(16)

Let S = S × H (D). Then, in view of (16) and of remarks on the supports of Pζ and Pζ,α , we have that the set S is a minimal set such that Pζ (S) = 1. The theorem is proved.  4 Proof of the main theorem Theorem 6 is a consequence of Theorems 7, 8 and of the Mergelyan theorem [11,20], on approximation of analytic functions by polynomials. Proof By the Mergelyan theorem, there exist polynomials p(s) and q(s) such that ! ! ε ! ! sup ! f (s) − e p(s) ! < 2 s∈K and sup | f 1 (s) − q(s)| < s∈K 1

(17)

ε . 2

(18)

Define a set G ⊂ H 2 (D) by 

! ! ε ε ! ! G = (g1 , g2 ) ∈ H 2 (D) : sup !g1 (s) − e p(s) ! < , sup |g2 (s) − q(s)| < . 2 s∈K 1 2 s∈K

Then G is an open set of the space H 2 (D). Moreover, since e p(s) , q(s) , by Theorem 8, is an element of the support of Pζ , using Theorem 7 and an equivalent of the weak convergence of probability measures in terms of open sets, Theorem 2.1 of [2], we have that lim inf N →∞

  1 # 0 ≤ k ≤ N : ζ (s + ikh, α) ∈ G ≥ Pζ (G) > 0. N +1

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E. Buivydas, A. Laurinˇcikas

This and the definition of the set G yield the inequality  ! ! ε 1 ! ! lim inf # 0 ≤ k ≤ N : sup !ζ (s + ikh) − e p(s) ! < , N →∞ N + 1 2 s∈K ε sup |ζ (s + ikh, α) − q(s)| < > 0. 2 s∈K 1

(19)

Clearly, in view of (17) and (18),  0 ≤ k ≤ N : sup |ζ (s + ikh) − f (s)| < ε, s∈K sup |ζ (s + ikh, α) − f 1 (s)| < ε  ! ! ε ! ! ⊃ 0 ≤ k ≤ N : sup !ζ (s + ikh) − e p(s) ! < , 2 s∈K ε sup |ζ (s + ikh, α) − q(s)| < . 2 s∈K 1 s∈K 1

Combining this with (19) gives the assertion of the theorem.



References 1. Bagchi, B.: The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series. Ph. D. Thesis, Indian Statistical Institute, Calcutta (1981) 2. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) 3. Bitar, K.M., Khari, N.N., Ren, H.C.: Path integrals and Voronin’s theorem on the universality of the Riemann zeta-function. Ann. Phys. 211, 172–196 (1991) 4. Gel’fond, A.O.: On the algebraic independence of transcendental numbers of certain classes. Uspehi Matem. Nauk (N. S.) 4, 5(33), 14–48 (1949) (in Russian) 5. Gonek, S.M.: Analytic properties of zeta and L-functions. Ph. D. Thesis, University of Michigan (1979) 6. Heyer, H.: Probability Measures on Locally Compact Groups. Springer, Berlin (1977) 7. Laurinˇcikas, A.: Limit Theorems for the Riemann Zeta-Function. Kluwer Academic Publishers, Dordrecht (1996) 8. Laurinˇcikas, A.: On the joint “universality” of Hurwitz zeta-functions. Šiauliai Math. Semin. 3(11), 169–187 (2008) 9. Laurinˇcikas, A., Garunkštis, R.: The Lerch Zeta-Function. Kluwer Academic Publishers, Dordrecht (2002) 10. Laurinˇcikas, A., Macaitien˙e, R.: The discrete universality of the periodic Hurwitz zeta-function. Integral Transf. Spec. Funct. 20(9), 673–686 (2009) 11. Mergelyan, S.N.: Uniform approximations to functions of a complex variable. Usp. Matem. Nauk 7, 31–122 (1952). (in Russian) 12. Mishou, H.: The joint value distribution of the Riemann zeta-function and Hurwitz zeta-functions. Lith. Math. J. 47(1), 32–47 (2007) 13. Montgomery, H.L.: Topics in Multiplicative Number Theory. Springer, Berlin (1971) 14. Nesterenko, Y.V.: Modular functions and transcendence questions, Mat. Sb. 187(9), 65–96 (1996) (in Russian) ≡ Sb. Math., 187(9), 1319–1348 (1996) 15. Reich, A.: Wertverteilung von Zetafunktionen. Arch. Math. 34, 440–451 (1980) 16. Sander, J., Steuding, J.: Joint “universality” for sums and products of Dirichlet L-functions. Analysis (Munich) 26(3), 295–312 (2006)

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A discrete version of the Mishou theorem 17. Steuding, J.: Value-Distribution of L-functions. Lecture Notes in Math, vol. 1877. Springer, Berlin (2007) 18. Voronin, S.M.: Theorem on the “universality” of the Riemann zeta-function, Izv. Akad. Nauk SSRS, Ser. Matem. 39, 475–486 (1975) (in Russian) ≡ Math. USSR Izv. 9, 443–453 (1975) 19. Voronin, S.M.: Analytic properties of Dirichlet generating functions of arithmetic objects. Thesis of doctor fiz.-mat. nauk, Steklov Math. Inst, Moscow (1977) (in Russian) 20. Walsh, J.L.: Interpolation and Approximation by Rational Functions on the Complex Domain, vol. 20. Amer. Math. Soc. Colloq. Publ. (1960)

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