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Article

Joint Universality of the Zeta-Functions of Cusp Forms

by
Renata Macaitienė
Regional Development Institute of Šiauliai Academy, Vilnius University, P. Višinskio Str. 25, LT-76351 Šiauliai, Lithuania
Mathematics 2021, 9(17), 2161; https://doi.org/10.3390/math9172161
Submission received: 26 July 2021 / Revised: 30 August 2021 / Accepted: 31 August 2021 / Published: 4 September 2021

Abstract

:
Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ ( s , F ) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts ( ζ ( s + i h 1 τ , F ) , , ζ ( s + i h r τ , F ) ) is proved. Here, h 1 , , h r are algebraic numbers linearly independent over the field of rational numbers.

1. Introduction

The series of the types
m = 1 a m m s and m = 1 a m e λ m s , s = σ + i t ,
where { λ m } is a nondecreasing sequence of real numbers and lim m λ m = + are called Dirichlet series. The majority of zeta-functions are meromorphic functions in some half-plane defined by Dirichlet series having a certain arithmetic sense. The most important of zeta-functions is the Riemann zeta-function
ζ ( s ) = m = 1 1 m s , σ > 1 .
In [1], Voronin discovered a very interesting and important property of ζ ( s ) to approximate a wide class of analytic functions by shifts ζ ( s + i τ ) , τ R , and called it universality. Later, it turned out that some other zeta-functions also are universal in the Voronin sense. This paper is devoted to the universality of zeta-functions of certain cusp forms.
Let
S L ( 2 , Z ) = a b c d : a , b , c , d Z , a d b c = 1
be the full modular group. If the function F ( z ) is holomorphic in the upper half-plane Im z > 0 , and for all elements of S L ( 2 , Z ) with some κ 2 N satisfies the functional equation
F a z + b c z + d   = ( c z + d ) κ F ( z ) ,
where F ( z ) is called a modular form of weight κ for the full modular group. Then, F ( z ) has Fourier series expansion
F ( z ) = m = c ( m ) e 2 π i m z .
If c ( m ) = 0 for all m 0 , then F ( z ) is a cusp form of weight κ . The corresponding zeta-function (or L-function) ζ ( s , F ) is defined for σ > κ + 1 2 by the Dirichlet series
ζ ( s , F ) = m = 1 c ( m ) m s ,
and has the analytic continuation to an entire function. Additionally, we suppose that F ( z ) is a simultaneous eigenfunction of all Hecke operators T m
T m F ( z ) = m κ 1 a , d > 0 a d = m 1 d κ b ( mod d ) F a z + b d , m N .
In this case, c ( 1 ) 0 ; therefore, the form F ( z ) can be normalized, and thus, we may suppose that c ( 1 ) = 1 .
Now, we suppose that F ( z ) is a normalized Hecke-eigen cusp form of weight κ for the full modular group. Then, the zeta-function ζ ( s , F ) can be written, for σ > κ + 1 2 , as a product over primes
ζ ( s , F ) = p 1 α ( p ) p s 1 1 β ( p ) p s 1 ,
where α ( p ) and β ( p ) are conjugate complex numbers satisfying the equality α ( p ) + β ( p ) = c ( p ) .
In the paper [2], the universality of the function ζ ( s , F ) was proved. Let D κ =   s C : κ 2 < σ < κ + 1 2 , K F be the class of compact subsets of the strip D κ with connected complements, and H 0 , F ( K ) , K K F the class of continuous nonvanishing functions on K that are analytic in the interior of K. Moreover, let meas A denote the Lebesgue measure of a measurable set A R . Then, in [2], the following theorem was obtained.
Theorem 1.
Suppose that K K F and f ( s ) H 0 , F ( K ) . Then, for every ε > 0 ,
lim inf T 1 T meas τ [ 0 , T ] : sup s K | ζ ( s + i τ , F ) f ( s ) | < ε   > 0 .
Theorem 1 shows that there are infinitely many shifts ζ ( s + i τ , F ) approximating a given function f ( s ) H 0 , F . In the shifts ζ ( s + i τ , F ) of Theorem 1, τ takes arbitrary real values; therefore, the theorem is of continuous type. Further, discrete universality theorems for the function ζ ( s , F ) are known. In [3,4], the discrete universality theorems with shifts ζ ( s + i k h , F ) , k N , h > 0 being a fixed number, were proved. Denote by H ( D κ ) the space of analytic on D κ functions endowed with the topology of uniform convergence on compacta. The paper [5] is devoted to the universality for compositions Φ ( ζ ( s , F ) ) with certain operators Φ : H ( D κ ) H ( D κ ) . The results of the latter paper were applied in [6] for the functional independence of the compositions Φ ( ζ ( s , F ) ) .
Let, for a fixed l N ,
Γ 0 ( l ) =   a b c d     S L ( 2 , Z ) : c 0 ( mod l )
denote the Hecke subgroup of the group S L ( 2 , Z ) . If F ( z ) satisfies (1) for all elements of Γ 0 ( l ) , then F ( z ) is called a cusp form of weight κ and level l. The form F ( z ) is called a new form if it is not a cusp form of level l 1 l . In [7], a universality theorem was obtained for zeta-functions of new forms.
The universality theorem of [2] was generalized in [8] for shifts ζ ( s + i φ ( τ ) , F ) with differentiable function φ ( τ ) satisfying the estimates ( φ ( τ ) ) 1 = o ( τ ) and φ ( 2 τ ) max τ t 2 τ ( φ ( t ) ) 1 τ as τ . The discrete version of results of [8] is given in [9]. In [10], the shifts ζ ( s + i γ k , F ) , where { γ k : k N } is the sequence of nontrivial zeros of ζ ( s ) , are used.
The joint universality of zeta- and L-functions is a more complicated problem of analytic number theory. In this case, a collection of analytic functions are simultaneously approximated by a collection of shifts of zeta-functions. The first result in this direction also belongs to Voronin. He considered [11] the functional independence of Dirichlet L-functions L ( s , χ ) with pairwise nonequivalent Dirichlet characters χ and, for this, he obtained their joint universality. The paper [12] is devoted to the joint universality for zeta-functions of new forms twisted by Dirichlet characters, i.e., for the functions
m = 1 c ( m ) χ ( m ) m s , σ > κ + 1 2 ,
with pairwise nonequivalent Dirichlet characters χ 1 , , χ r .
Joint universality theorems with generalized shifts ζ ( s + i φ j ( k ) , F ) , j = 1 , , r , with some differentiable functions φ j ( τ ) can be found in [13]. Continuous and discrete joint universality theorems for more general zeta-functions are given in [14,15,16].
Our aim is to obtain a joint universality theorem for zeta-functions of normalized Hecke-eigen cusp forms by using different shifts. The first of the denseness results for shifts of a universal function were discussed in [17].
The main result of the paper is the following statement.
Theorem 2.
Suppose that h 1 , , h r are real algebraic numbers linearly independent over the field of rational numbers Q . For j = 1 , , r , let K j K F and f j ( s ) H 0 , F ( K j ) . Then, for every ε > 0 ,
lim inf T 1 T meas τ [ 0 , T ] : sup 1 j r sup s K j | ζ ( s + i h j τ , F ) f j ( s ) | < ε   > 0 .
Moreover “lim inf” can be replaced by “lim” for all but at most countably many ε > 0 .
For the proof of Theorem 2, we will apply the probabilistic approach based on a limit theorem in the space of analytic functions.

2. Mean Square Estimates

Recall the metric in the space H ( D κ ) . Let { K l : l N } D κ be a sequence of compact subsets such that
D κ = l = 1 K l ,
K l K l + 1 for l N , and if K D κ is a compact, then K K l for some l. For example, we can take K l closed rectangles. Then
ρ ( g 1 , g 2 ) = l = 1 2 l sup s K l g 1 ( s ) g 2 ( s ) 1 + sup s K l g 1 ( s ) g 2 ( s ) , g 1 , g 2 H ( D κ ) ,
is a metric in H ( D κ ) inducing the topology of uniform convergence on compacta.
Let
H r ( D κ ) =   ( H ( D κ ) × × H ( D κ ) r .
For g ̲ j = ( g j 1 , , g j r ) H r ( D κ ) , j = 1 , 2 , define
ρ ̲ ( g ̲ 1 , g ̲ 2 ) = max 1 j r ρ ( g 1 j , g 2 j ) .
Then, ρ ̲ is a metric in H r ( D κ ) inducing the product topology.
Let θ > 1 2 be a fixed number, and
v n ( m ) = exp m n θ , m , n N .
Then, the series
ζ n ( s , F ) = m = 1 c ( m ) v n ( m ) m s ,
in view of the estimate
c ( m ) m κ 1 2 + ε ,
is absolutely convergent in every fixed half plane σ > σ ^ . However, for our aim, this convergence is sufficient only for σ > κ 2 .
For brevity, let h ̲ = ( h 1 , , h r ) ,
ζ ̲ ( s + i h ̲ τ , F ) =   ζ ( s + i h 1 τ , F ) , , ζ ( s + i h r τ , F )
and
ζ ̲ n ( s + i h ̲ τ , F ) =   ζ n ( s + i h 1 τ , F ) , , ζ n ( s + i h r τ , F ) .
Lemma 1.
For all h ̲ ,
lim n lim sup T 1 T 0 T ρ ̲ ζ ̲ ( s + i h ̲ τ , F ) , ζ ̲ n ( s + i h ̲ τ , F ) d τ = 0 .
Proof. 
By the definitions of the metrics ρ and ρ ̲ , it suffices to show that, for every h R and compact set K D κ ,
lim n lim sup T 1 T 0 T sup s K ζ ( s + i h τ , F ) , ζ n ( s + i h τ , F ) d τ = 0 .
It is well known that for fixed κ 2 < σ < κ + 1 2 ,
T T ζ ( σ + i t , F ) 2 d t σ T ,
where σ means that the implied constant depends on σ . Therefore,
T T ζ ( σ + i h t , F ) 2 d t σ , h T ,
and, for v R ,
1 T 0 T ζ ( σ + i h τ + i v , F ) 2 d v σ , h 1 + | v | .
Let
l n ( s ) = z θ Γ z θ n z ,
where Γ ( z ) denotes the Euler gamma-function and θ is a number from the definition of v n ( m ) . Using the Mellin formula
1 2 π i β i β + i Γ ( s ) α s d s = e α , α , β > 0 ,
we find that
exp m n θ   = 1 2 π i θ i θ + i 1 θ Γ 1 θ m n s d s .
Therefore, in virtue of the definition of the function v n ( m ) , we obtain that, for σ > κ 2 ,
ζ n ( s , F ) = 1 2 π i m = 1 c ( m ) m s θ i θ + i z θ Γ z θ m n z d z z
= 1 2 π i θ i θ + i l n ( z ) z m = 1 c ( m ) m s + z d z
= 1 2 π i θ i θ + i ζ ( s + z , F ) l n ( z ) d z z .
Let K D κ be a fixed compact set. Then, there exists ε > 0 such that, for all s = σ + i t K , the inequalities κ 2 + 2 ε < σ < κ + 1 2 ε are satisfied. We take, for such σ ,
θ 1 = κ 2 + ε σ .
Then, θ 1 < 0 . Therefore, by the residue theorem and (3),
ζ n ( s , F ) ζ ( s , F ) = 1 2 π i θ 1 i θ 1 + i ζ ( s + z , F ) l n ( z ) d z z .
Hence, for all s K ,
ζ ( s + i h τ , F ) ζ n ( s + i h τ , F ) = 1 2 π i ζ κ 2 + ε + i t + i h τ + i v , F l n κ 2 + ε σ + i v κ 2 + ε σ + i v d v = 1 2 π i ζ κ 2 + ε + i h τ + i v , F l n κ 2 + ε s + i v κ 2 + ε s + i v d v 1 2 π i ζ κ 2 + ε + i h τ + i v , F sup s K l n κ 2 + ε s + i v κ 2 + ε s + i v d v .
Thus, in view of (2),
1 T 0 sup s K ζ ( s + i h τ , F ) ζ n ( s + i h τ , F ) d τ
1 T 0 ζ κ 2 + ε + i h τ + i v 2 d τ 1 / 2 sup s K l n κ 2 + ε s + i v κ 2 + ε s + i v d v
ε , h , K n ε ( 1 + | v | ) exp { c 1 | v | } d v ε , h , K n ε
Here, we used the estimate
Γ 1 θ κ 2 + ε s + i v     exp c θ | v t |   κ   exp { c 1 | v | } , c 1 > 0 .
Estimate (4) proves the lemma. □
Let P be the set of all prime numbers, and γ p = { s C : | s | = 1 } for all p P . Define the set
Ω = p P γ p .
Then, the torus Ω with product topology and pointwise multiplication is a compact topological Abelian group. Therefore, on ( Ω , B ( Ω ) ) ( B ( X ) is the Borel σ -field of the space X ), the probability Haar measure m H can be defined. Moreover, let
Ω ̲ = Ω 1 × × Ω r ,
where Ω j = Ω for all j = 1 , , r . Once again, Ω ̲ is a compact topological Abelian group. Therefore, on ( Ω ̲ , B ( Ω ̲ ) ) the probability Haar measure m ̲ H exists. This gives the probability space ( Ω ̲ , B ( Ω ̲ ) , m ̲ H ) . Denote by m j H the Haar measure on ( Ω j , B ( Ω j ) ) , j = 1 , , r . Then, m ̲ H is the product of the measures m 1 H , , m r H . Now, denote by ω ̲ = ( ω 1 , , ω r ) the elements of Ω ̲ , where ω j Ω j , j = 1 , , r . Let ω j ( p ) be the pth component of an element ω j Ω j , j = 1 , , r , p P . Extend elements ω j ( p ) to the set N by the formula
ω j ( m ) = p l m p l + 1 m ω j l ( p ) , m N ,
and define H ( D κ ) -valued random element
ζ ( s , ω j , F ) = m = 1 c ( m ) ω j ( m ) m s , j = 1 , , r .
The later series is uniformly convergent on compact subsets of D κ for almost all ω j . Moreover, for fixed σ κ 2 , κ + 1 2
T T ζ ( s + i t , ω j , F ) 2 d t σ T
for almost all ω j , j = 1 , , r [18]. Define one more series
ζ n ( s , ω j , F ) = m = 1 c ( m ) ω j ( m ) v n ( m ) m s , j = 1 , , r ,
which also, as ζ n ( s , F ) , are absolutely convergent for σ > κ 2 . Let
ζ ̲ ( s + i h ̲ τ , ω ̲ , F ) =   ζ ( s + i h 1 τ , ω 1 , F ) , , ζ ( s + i h r τ , ω 1 , F )
and
ζ ̲ n ( s + i h ̲ τ , ω ̲ , F ) =   ζ n ( s + i h 1 τ , ω 1 , F ) , , ζ n ( s + i h r τ , ω r , F ) .
Then, repeating the proof of Lemma 1 and using estimate (5), we arrive to the following statement.
Lemma 2.
For all h ̲ and almost all ω ̲ ,
lim n lim sup T 1 T 0 T ρ ̲ ζ ̲ ( s + i h ̲ τ , ω ̲ , F ) , ζ ̲ n ( s + i h ̲ τ , ω ̲ , F ) d τ = 0 .

3. Limit Theorems

On the probability space ( Ω ̲ , B ( Ω ̲ ) , m ̲ H ) , define H ( D κ ) -valued random element
ζ ̲ ( s , ω ̲ , F ) =   ζ ( s , ω 1 , F ) , , ζ ( s , ω 1 , F )
and denote by P ζ ̲ , F its distribution, i.e.,
P ζ ̲ , F ( A ) = m ̲ H ω ̲ Ω ̲ : ζ ̲ ( s , ω ̲ , F ) A , A B ( H r ( D κ ) ) .
Theorem 3.
Suppose that h 1 , , h r are real algebraic numbers linearly independent over Q , and
P T , F ( A ) = d e f 1 T meas τ [ 0 , T ] : ζ ̲ ( s + i h ̲ τ , F ) A , A B ( H r ( D κ ) ) .
Then, P T , F converges weakly to P ζ ̲ , F as T .
We divide the proof of Theorem 3 into several lemmas.
Lemma 3.
Suppose that λ 1 , , λ r are algebraic numbers such that the system log λ 1 , , log λ r is linearly independent over Q . Then, for arbitrary algebraic numbers β 0 , β 1 , , β r that are not all zeros, the inequality
| β 0 + β 1 log λ 1 + + β r log λ r |   > h c
holds. Here, h denotes the height of the numbers β 0 , β 1 , , β r , and c is an effective constant depending on r, λ 1 , , λ r and maximum of degrees of the numbers β 0 , β 1 , , β r .
The lemma is a Baker result on linear forms of logarithm; see, for example, ref. [19]. For A B ( Ω ̲ ) , define
Q T ( A ) = 1 T meas τ [ 0 , T ] : p i h 1 τ : p P , , p i h r τ : p P   A .
Lemma 4.
Let λ 1 , , λ r be the same as in Theorem 3. Then, Q T converges weakly to the Haar measure m ̲ H as T .
Proof. 
We apply the Fourier transform method. Denote by g T ( k ̲ 1 , , k ̲ r ) , k ̲ j = { k p j : k p j Z , p P } , j = 1 , , r the Fourier transform of Q T . By the definition of Q T , we have
g T ( k ̲ 1 , , k ̲ r ) = Ω ̲ j = 1 r * p P ω j k p j ( p ) d Q T 1 T 0 T exp i τ j = 1 k * p P h j k p j log p d τ ,
where the star shows that only a finite number of integers k p j are not zero. Obviously,
g T ( 0 ̲ , , 0 ̲ ) = 1 .
Now, suppose that ( k ̲ 1 , , k ̲ r ) ( 0 ̲ , , 0 ̲ ) . Then, there exists a prime number p such that k p j 0 for some j. Therefore,
β p = d e f j = 1 r h j k p j 0
because the numbers h 1 , , h r are linearly independent over Q. Thus, in view of Lemma 3,
B k ̲ 1 , , k ̲ r = d e f j = 1 k * p P h j k p j log p = * p P β p log p 0 .
This and (6) imply
g T ( k ̲ 1 , , k ̲ r ) = 1 exp i T B k ̲ 1 , , k ̲ r i T B k ̲ 1 , , k ̲ r .
Therefore, by (7),
lim T g T ( k ̲ 1 , , k ̲ r ) = d e f 1 if ( k ̲ 1 , , k ̲ r ) = ( 0 ̲ , , 0 ̲ ) , 0 if ( k ̲ 1 , , k ̲ r ) ( 0 ̲ , , 0 ̲ ) ,
and this proves the lemma. □
For A B ( H r ( D κ ) ) , define
P T , n , F ( A ) = 1 T meas τ [ 0 , T ] : ζ ̲ n ( s + i h ̲ τ , F ) A
and
P T , n , Ω ̲ , F ( A ) = 1 T meas τ [ 0 , T ] : ζ ̲ n ( s + i h ̲ τ , ω ̲ , F ) A .
Moreover, let the mapping u n : Ω ̲ H r ( D κ ) be given by
u n , F ( ω ̲ ) = ζ ̲ n ( s , ω ̲ , F ) ,
and V n , F = m ̲ H u n , F 1 , where
V n , F ( A ) = m ̲ H u n , F 1 A , A B ( H r ( D κ ) ) .
Since the series for ζ n ( s , ω j , F ) are absolutely convergent for σ > κ 2 , the mapping u n , F is continuous. Moreover, by the definitions of Q T and P T , n , F , we have P T , n , F = Q T u n , F 1 . This equality, continuity of u n , F , Lemma 4, the well-known properties of weak convergence, and the invariance of the Haar measure m ̲ H lead to the following lemma.
Lemma 5.
Let h 1 , , h r be the same as Theorem 3. Then, P T , n , F and P T , n , Ω ̲ , F both converge weakly to the measure V n , F as T .
Additionally to P T , F , define
P T , Ω ̲ , F ( A ) = 1 T meas τ [ 0 , T ] : ζ ̲ ( s + i τ , ω ̲ , F ) A , A B ( H r ( D κ ) ) .
Lemma 6.
Let h 1 , , h r be the same as Theorem 3. Then, on ( H r ( D κ ) , B ( H r ( D κ ) ) ) , there exists a probability measure P F such that P T , F and P T , Ω ̲ , F both converge weakly to P F as T .
Proof. 
Since the series for ζ n ( s , F ) is absolutely convergent, by a standard way it follows—see, for example [14,18]—that the sequence { V n , F : m N } is tight, i.e., for every ε > 0 , there exists a compact set K H r ( D κ ) ) such that
V n , F ( K ) > 1 ε
for all n N . Hence, by the Prokhorov theorem, see [20], the sequence { V n , F } is relatively compact, i.e., each of its subsequences contains a subsequence { V n k , F } such that V n k , F converges weakly to a certain probability measure P F on ( H r ( D κ ) , B ( H r ( D κ ) ) ) as k .
Let ξ T be a random variable defined on a certain probability space with measure ν and uniformly distributed on [ 0 , T ] . Define the H r ( D κ ) -valued random element
X ̲ T , n , F = X ̲ T , n , F ( s ) = ζ ̲ n ( s + i h ̲ ξ T , F )
and denote by X ̲ n , F = X ̲ n , F ( s ) the H r ( D κ ) -valued random element having the distribution V n , F . Then, by Lemma 5, we have
X ̲ T , n , F T D X ̲ n , F ,
where T D means the convergence in distribution. Moreover, since V n k , F converges weakly to P F , the relation
X ̲ n k , F k D P F
is true. Let
X ̲ T , F = X ̲ T , F ( s ) = ζ ̲ ( s + i h ̲ ξ T , F ) .
Then, using Lemma 1, we find that for every ε > 0 ,
lim n lim sup T ν ρ ̲ X ̲ T , F , X ̲ T , n , F   ε lim n lim sup T 1 ε T 0 T ρ ̲ ζ ̲ ( s + i h ̲ τ , F ) , ζ ̲ n ( s + i h ̲ τ , F ) d τ = 0 .
The later equality together with (8) and (9), and Theorem 4.2 of [20] lead to the relation
X T , F T D P F .
This proves that P T , F converges weakly to P F as T .
The relation (10) shows that the limit measure P F is independent of the subsequence { n k } . Therefore, we have
X ̲ n , F n D P F .
Define the H r ( D κ ) -valued random elements
X ̲ T , n , Ω ̲ , F = X ̲ T , n , Ω ̲ , F ( s ) = ζ ̲ n ( s + i h ̲ ξ T , ω ̲ , F )
an
X ̲ T , Ω ̲ , F = X ̲ T , Ω ̲ , F ( s ) = ζ ̲ ( s + i h ̲ ξ T , ω ̲ , F ) .
Then, repeating the above arguments using Lemmas 2 and 5, and relation (11), we obtain that
X T , n , F T D P F ,
and this is equivalent to weak convergence of P T , Ω ̲ , F to P F as T . The lemma is proved. □
To prove Theorem 3, it remains to show that P F = P ζ ̲ , F . For this, we will apply some elements of the ergodic theory. For brevity, let
h ̲ τ =   p i h 1 τ : p P , , p i h r τ : p P , τ R .
Define the transformation of Ω ̲
φ τ ( ω ̲ ) = h ̲ τ ω ̲ , ω ̲ Ω ̲ .
Since the Haar measure m ̲ H is invariant, the transformation φ τ is measure-preserving and { φ τ : τ R } is a one-parameter group. A set A B ( Ω ̲ ) is called invariant with respect to the group { φ τ } if the sets A and φ τ ( A ) , τ R , differ one from another at most by a set of m ̲ H -measure zero.
Lemma 7.
Let h 1 , , h r be the same as Theorem 3. Then, the group { φ τ } is ergodic, i.e., the σ-field of invariant sets consists of sets having m ̲ H -measure 1 or 0.
Proof. 
The characters χ of the group Ω are of the form
χ ( ω ̲ ) = j = 1 r * p P ω j k p j ( p ) .
This fact already was used in the proof of Lemma 4. Let A be an arbitrary invariant set, I A its indicator function, and χ be a nontrivial character. Preserving the notation of the proof of Lemma 4, we have ( k ̲ 1 , , k ̲ r ) ( 0 ̲ , , 0 ̲ ) and B k ̲ 1 , , k ̲ r 0 . Therefore, there exists τ 0 R such that
χ ( h ̲ τ ) = exp i τ 0 B k ̲ 1 , , k ̲ r   1 .
Moreover, in view of the invariance of A, we have
I A ( h ̲ τ 0 ω ̲ ) = I A ( ω ̲ )
for almost all ω ̲ Ω ̲ . Denote by I ^ A the Fourier transform of I A . Then, by (13),
I ^ A ( χ ) = χ ( h ̲ τ 0 ) Ω ̲ I A ( h ̲ τ 0 ω ̲ ) χ ( ω ̲ ) d m ̲ H = χ ( h ̲ τ 0 ) I ^ A ( χ ) .
This and (12) show that
I ^ A ( χ ) = 0 .
Now, let χ 0 denote the trivial character of Ω ̲ , and suppose that I ^ A ( χ 0 ) = α . Then, in view of (14), we find that
I ^ A ( χ ) = α Ω ̲ χ ( ω ̲ ) d m ̲ H = α ^ ( χ ) .
Hence, I A ( ω ̲ ) = α for almost all ω ̲ Ω ̲ . Since I A is the indicator function, I A ( ω ̲ ) = 1 or I A ( ω ̲ ) = 0 for almost all ω ̲ . Thus, m ̲ H ( A ) = 1 or m ̲ H ( A ) = 0 , and the lemma is proved. □
Proof of Theorem 3.
We have mentioned that it suffices to show that P F = P ζ ̲ , F . By Lemma 6 and the equivalent of weak convergence in terms of continuity sets, we have
lim T P T , Ω ̲ , F ( A ) = P F ( A )
for a continuity set A of the measure P F , i.e., P F ( A ) = 0 , where A is the boundary of A. On the probability space ( Ω ̲ , B ( Ω ̲ ) , m ̲ H ) , define the random variable
ξ ( ω ̲ ) =   1 if ζ ̲ ( s , ω ̲ , F ) A , 0 otherwise .
Lemma 7 implies the ergodicity of the random process ξ ( φ τ ( ω ̲ ) ) . Therefore, by the classical Birkhoff–Khintchine ergodic theorem, see, for example [21],
lim T 1 T 0 T ξ ( φ τ ( ω ̲ ) ) d τ = E ξ = P ζ ̲ , F ( A ) ,
where E ξ is the expectation of ξ .
However, by the definitions of φ τ and ξ ,
1 T 0 T ξ ( φ τ ( ω ̲ ) ) d τ = 1 T meas τ [ 0 , T ] : ζ ̲ ( s + i h ̲ τ , ω ̲ , F ) A   = P T , Ω ̲ , F ( A ) .
This and (16) show that
lim T P T , Ω ̲ , F ( A ) = P ζ ̲ , F ( A ) .
Therefore, by (15), we obtain that P F ( A ) = P ζ ̲ , F ( A ) for all continuity sets A of P F ( A ) . Hence, P F = P ζ ̲ , F , and the theorem is proved. □

4. Proof of Theorem 2

Recall that the support of the measure P ζ ̲ , F is a minimal closed set S F H r ( D κ ) such that P ζ ̲ , F ( S F ) = 1 .
Lemma 8.
The support of the measure P ζ ̲ , F is the set g H ( D κ ) : g ( s ) 0 o r g ( s ) 0 r .
Proof. 
Since the space H r ( D κ ) is separable, we have [20],
B H r ( D κ )   =   ( B ( H ( D κ ) ) × × B ( H ( D κ ) ) r .
Therefore, it suffices to consider the measure P ζ ̲ , F on the rectangular sets
A = A 1 × × A r , A 1 , , A r H ( D κ ) .
Let ζ ( s , ω , F ) be the H ( D κ ) -valued random element defined on the probability space ( Ω , B ( Ω ) , m H ) , where m H is the Haar measure. Then, it is known [10] that the support of the distribution of ζ ( s , ω , F ) is the set { g H ( D κ ) : g ( s ) 0 or g ( s ) 0 } . Thus, the same set is the support of the distributions of ζ ( s , ω j , F ) , j = 1 , , r . Since the measure m ̲ H is the product of the measures m j H , j = 1 , , r , we have
m ̲ H ω ̲ Ω ̲ : ζ ̲ ( s , ω ̲ , F ) A   = j = 1 r m j H ω j Ω j : ζ ( s , ω j , F ) A j .
This equality, the minimality of the support, and the support of the distributions of ζ ( s , ω j , F ) prove the lemma. □
Proof of Theorem 2.
By the Mergelyan theorem on the approximation of analytic functions by polynomials [22], there exist polynomials p 1 ( s ) , , p r ( s ) such that
sup 1 j r sup s K j f j ( s ) e p j ( s )   < ε 2 .
Define the set
G ε =   ( g 1 , , g r ) H r ( D κ ) : sup 1 j r sup s K j g j ( s ) e p j ( s )   < ε 2 .
In view of Lemma 8, the set G ε is an open neighborhood of an element ( e p 1 ( s ) , , e p r ( s ) ) in support of the measure P ζ ̲ , F . Hence,
P ζ ̲ , F ( G ε ) > 0 .
This, Theorem 3 and the equivalent of weak convergence in terms of open sets, and the definitions of P T , F and G ε prove the theorem with “lim inf”. Define one more set
G ^ ε =   ( g 1 , , g r ) H r ( D κ ) : sup 1 j r sup s K j g j ( s ) f j ( s )   < ε ,
There G ^ ε 1 G ^ ε 2 = for ε 1 ε 2 . This shows that P ζ ̲ , F G ^ ε = 0 for all but, for those countable, many ε > 0 . Moreover, (17) and (18) imply that P ζ ̲ , F G ^ ε > 0 . This, Theorem 3 and the equivalent of weak convergence of probability measures in terms of continuity sets, and the definitions of P T , F and G ^ ε prove the theorem with “lim”. □

Funding

This research received no external funding.

Conflicts of Interest

The author declares no conflict of interest.

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Macaitienė, R. Joint Universality of the Zeta-Functions of Cusp Forms. Mathematics 2021, 9, 2161. https://doi.org/10.3390/math9172161

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Macaitienė, Renata. 2021. "Joint Universality of the Zeta-Functions of Cusp Forms" Mathematics 9, no. 17: 2161. https://doi.org/10.3390/math9172161

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Macaitienė, R. (2021). Joint Universality of the Zeta-Functions of Cusp Forms. Mathematics, 9(17), 2161. https://doi.org/10.3390/math9172161

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