Abstract
In this paper, we define the Dirichlet series \( \zeta_{u_T j} (s)\), \( j = 1, \dots, r\), absolutely converging in the half-plane \( \operatorname{Re} s> 1/2 \) and prove that the set of shifts \( (\zeta_{u_T 1} (s + ia_1 \tau), \dots, \zeta_{u_T r} (s + ia_r \tau)) \) approximating a given set of analytic functions has a positive density on the interval \( [T, T + H]\), \( H = o (T) \) as \( T \to \infty\). Here \( a_1, \dots, a_r \in \mathbb{R} \) are algebraic numbers linearly independent over \( \mathbb{Q} \) and \( u_T \to \infty \) as \( T \to \infty\).
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1. Introduction
Continuing the research of Bohr and Courant on the density of the set of values of the Riemann zeta function \( \zeta (s)\), \( s = \sigma + it\), S. M. Voronin discovered the universality property of this function in [1]. Let \( 0 <r <1/4\). He proved that if the function \( f (s) \) is analytic in a disk \( | s | <r\), and it is continuous and has no zeros up to the boundary of this disk, then, for every \( \varepsilon> 0\), there exists a \( \tau = \tau (\varepsilon) \in \mathbb{R} \) such that
Since the space of analytic functions is infinite-dimensional, Voronin’s theorem is an infinite-dimensional generalization of the result due to Bohr and Courant [2] that, for \( 1/2 <\sigma \le 1\), states that the set
is everywhere dense in \( \mathbb{C}\).
Voronin’s theorem has a more general form. Let \( D = \{s \in \mathbb{C}: 1/2 <\sigma <1 \}\), let \( \mathcal{K} \) be the class of compact subsets of the strip \( D \) having connected complements, and let \( H_0 (K)\), \( K \in \mathcal{K}\), be the class of continuous functions without zeros in \( K \) and analytic inside \( K\). In that case (see, for example, [3]) if \( K \in \mathcal{K} \) and \( f (s) \in H_0 (K)\), then, for every \( \varepsilon> 0\),
Here \( \operatorname{meas} A \) denotes the Lebesgue measure of a measurable set \( A \subset \mathbb{R}\). The last inequality shows that there exists infinitely many shifts \( \zeta (s + i \tau) \) approximating this function from the class \( H_0 (K)\).
Several modifications of Voronin’s theorem are known. Universality theorems on the approximation of analytic functions by generalized shifts \( \zeta (s + i \gamma (\tau)) \) with some function \( \gamma (\tau) \) are known [4], [5], and proofs were given for discrete universality theorems with shifts \( \zeta (s + ikh)\), \( h> 0\), \( k \in \mathbb{N} _0 = \mathbb{N} \cup \{0 \}\), [6], and \( \zeta (s + i \gamma (k)) \) [4], [7]-[9], or even with \( \zeta (s + i \gamma_k) \) [10] and \( \zeta (s + it_k) \) [11], where \( \{\gamma_k> 0 \} \) is the sequence of imaginary parts of nontrivial zeros of the function \( \zeta (s)\), and \( \{t_k \} \) is the sequence of Gram points. Also known are universality theorems with weights [12], [13]. In [14], “\( \liminf \)” in (1.1) was replaced by “\( \lim \)”.
More complicated are joint universality theorems when several analytic functions are approached simultaneously by shifts of the same zeta function. The first joint universality theorem was also obtained by Voronin for Dirichlet \( L \)-functions [15].
It is clear that, in joint universality theorems, the approximating shifts must be independent in some sense. So, in [15], shifts with pairwise nonequivalent Dirichlet characters were used. In [16], the following theorem was obtained.
Theorem 1.
Suppose that \( a_1, \dots, a_r \) are real algebraic numbers linearly independent over the field of rational numbers \( \mathbb{Q} \) and
where
Let \( K_j \in \mathcal{K}\) , and let \( f_j (s) \in H_0 (K_j)\) , \( j = 1, \dots, r\) . Then, for every \( \varepsilon> 0\) ,
Also, “ \( \liminf \) ” can be replaced by “ \( \lim \) ” with the exception of at most a countable set of values \( \varepsilon> 0\) .
The purpose of this paper is to replace the function \( \zeta (s) \) in Theorem 1 by an absolutely convergent Dirichlet series.
Let \( \theta_j> 1/2 \) be a fixed number, let \( u> 0\), and let
We define
Then these series absolutely converge in the plane \( \sigma> \sigma_0 \) for any fixed \( \sigma_0\). Inequality \( \theta_j> 1/2 \) is related to representation (3.2) and its application to the proof of Lemma 3, because the inequality \( \operatorname{Re} (s + z)> 1 \) must hold for \( s \in K\). Let us define another probability space. Let \( \mathcal{B} (\mathbb{X}) \) denote the Borel \( \sigma \)-field of the space \( \mathbb{X}\). We put
where \( \mathbb{P} \) is the set of all primes and \( \gamma_p = \{s \in \mathbb{C}: | s | = 1 \} \) for all \( p \in \mathbb{P}\). With the product topology and pointwise multiplication, the set \( \Omega \) is a compact topological Abelian group by Tikhonov’s theorem. Therefore, Haar probability measure can be defined on \( (\Omega, \mathcal{B} (\Omega))\). Further, let
where \( \Omega_j = \Omega \) for all \( j = 1, \dots, r\). Then \( \underline{\Omega} \) is again a compact topological group and there is a Haar measure \( m_H \) on \( (\underline{\Omega}, \mathcal{B} (\underline{\Omega} \mspace{1mu}))\). We obtain the probability space \( (\underline{\Omega}, \mathcal{B} (\underline{\Omega} \mspace{1mu}), m_H)\). Let \( \omega (p) \) be the \( p \)th component of an element \( \omega_j \in \Omega_j\), \( j = 1, \dots, r\), \( p \in \mathbb{P}\). Let \( H (D) \) denote the space of analytic functions in the strip \( D\), functions equipped with the topology of uniform convergence on compact sets. Let
and, on the probability space \( (\underline{\Omega}, \mathcal{B} (\underline{\Omega} \mspace{1mu}), m_H)\), we define the \( H ^ r (D) \)-valued random element
where
For almost all \( \omega_j\), this product converges uniformly on compact subsets of the strip \( D\). Then the following statement holds.
Theorem 2.
Let \( a_1, \dots, a_r \) and \( H \) be the same as in Theorem 1 ; let \( u_T \to \infty \) and \( u_T \ll \exp \{o (T) \} \) as \( T \to \infty\) . Let \( K_j \in \mathcal{K}\) ; let \( f_j (s) \in H_0 (K_j)\) , \( j = 1, \dots, r\) . Then, with the exception of at most a countable set of values \( \varepsilon_1> 0, \dots, \varepsilon_r> 0\) , the following limit exists:
Theorem 2 is derived by a slight modification of Theorem 1 and from the proximity of functions \( \zeta (s) \) and \( \zeta_{uj} (s) \) by using the method of characteristic functions. For absolutely convergent Dirichlet series, one-dimensional universality theorems were obtained in [17]-[20].
2. Joint Universality of the Function \( \zeta (s) \)
For brevity, let
and, for sets \( A \in \mathcal{B} (H ^ r (D))\), let
Lemma 1 [16].
Suppose that \( a_1, \dots, a_r\) , and let \( H \) be the same as in Theorem 1 . Then, as \( T \to \infty\) , \( P_{T, H} \) converges weakly to the distribution of the random element \( \underline{\zeta} (s, \underline{\omega} \mspace{1mu})\) , i.e., to the measure
In addition, the set
is the support of the measure \( P _{\underline{\zeta}}\) .
Lemma 2.
Suppose that \( a_1, \dots, a_r \) and \( H \) are the same as in Theorem 1 . Let \( K_j \in \mathcal{K} \) and \( f_j (s) \in H_0 (K_j)\) , \( j = 1, \dots, r\) . Then, with the exception of at most a countable set of values \( \varepsilon_1> 0, \dots, \varepsilon_r> 0\) , the following limit exists:
Proof.
Let the mapping \( h \colon H ^ r (D) \to \mathbb{R} ^ r \) be defined by the formula
Then \( h \) is continuous. Therefore, from the properties of the weak convergence of probability measures and from Lemma 1, we find that, as \( T \to \infty\), \( P_{T, H} h ^{- 1} \) converges weakly to \( P _{\underline{\zeta}} h ^{- 1}\), where
It is well known that the weak convergence of probability measures on \( (\mathbb{R} ^ r, \mathcal{B} (\mathbb{R} ^ r)) \) is equivalent to the weak convergence of distribution functions. Let us define the distribution functions
Then we see that, for \( T \to \infty\), \( F_{T, H} (\varepsilon_1, \dots, \varepsilon_r) \) converges weakly to \( F (\varepsilon_1, \dots, \varepsilon_r)\), i.e., converges for \( \varepsilon_1, \dots, \varepsilon_r\), such that the one-dimensional distribution functions \( F (\varepsilon_1, \infty, \dots, \infty), \dots, F (\infty, \dots, \infty, \varepsilon_r) \) are continuous. Since the distribution function has at most a countable set of points of discontinuity, the lemma will be proved if we show the positivity of \( F (\varepsilon_1, \dots, \varepsilon_r)\).
By Mergelyan’s theorem on the approximation of analytic functions [21], here exist polynomials \( p_1 (s), \dots, p_r (s) \) such that
By Lemma 1, the collection \( (\mathrm{e} ^{p_1 (s)}, \dots, \mathrm{e} ^{p_r (s)}) \) is an element of the support of the measure \( P _{\underline{\zeta}}\); hence
where
However, it follows from inequalities (2.1) that the set \( G _{\varepsilon_1, \dots, \varepsilon_r} \) lies in the set
which, together with (2.2), proves that \( F (\varepsilon_1, \dots, \varepsilon_r)> 0\).
3. Proximity of \( \zeta (s) \) and \( \zeta_{u_T} (s) \)
Let \( \theta> 1/2\), \( u> 0\),
Let us prove the following statement.
Lemma 3.
Suppose that \( a_1, \dots, a_r \) and \( H \) are the same as in Theorem 1 . Let \( u_T \to \infty \) and \( u_T \ll \exp \{o (T) \} \) as \( T \to \infty\) . Then, for every \( a \in \mathbb{R} \setminus \{0 \} \) and any compact set \( K \subset D\) ,
Proof.
It was found in [16] that, for a fixed \( \sigma \in (1 / 2,1) \) and for every \( \tau \in \mathbb{R}\),
For \( \zeta_{u_T} (s) \) in the strip \( D\), we have the following representation [3]:
Let \( K \subset D \) be a fixed compact set. Then there will be an \( \varepsilon> 0 \) such that, for every \( s = \sigma + it \in K\), the inequalities \( 1/2 + 2 \varepsilon \le \sigma \le 1- \varepsilon \) will hold. For such \( \sigma \) we have \( \widehat{\theta} = 1/2 + \varepsilon- \sigma <0\). Therefore, representation (3.2) and the residue theorem lead to the equality
After elementary transformations, this formula can be written as
The gamma function \( \Gamma (\sigma + it) \) satisfies the estimate
which is uniform in any strip \( \sigma_1 <\sigma <\sigma_2\). Hence
From this and estimate (3.1), we obtain
Similarly, we find that, for all \( s \in K\),
Therefore,
Since \( u_T \ll \exp \{\varepsilon_T T \} \) as \( \varepsilon_T \to 0\), in view of (3.4) and (3.3), we obtain the assertion of the lemma.
4. Proof of Theorem 2
Theorem 2 readily follows from Lemmas 2 and 3.
Proof of Theorem 2.
It suffices to show that the distribution function
as \( T \to \infty \) converges weakly to the distribution function \( F (\varepsilon_1, \dots, \varepsilon_r)\). To do this, we use the method of characteristic functions. Let \( \varphi_{T, H} (v_1, \dots, v_r)\), \( \widehat{\varphi} _{T, H} (v_1, \dots, v_r)\), and \( \varphi (v_1, \dots, v_r)\), \( v_1, \dots, v_r \in \mathbb{R}\), be the characteristic functions, respectively, of the distribution functions \( F_{T, H} (\varepsilon_1, \dots, \varepsilon_r)\), \( \widehat{F} _{T, H} (\varepsilon_1, \dots, \varepsilon_r)\), and \( F (\varepsilon_1, \dots, \varepsilon_r)\). It is well known that the weak convergence of distribution functions implies the convergence of the corresponding characteristic functions. Therefore, from the proof of Lemma 2, we obtain
uniformly over \( | v_j | \le C_j \) for any \( 0 <C_j <\infty\), \( j = 1, \dots, r\). From the definition of the characteristic function, we obtain
Thus, by virtue of the inequality \( | \mathrm{e} ^{iv}-1 | \le | v |\), \( v \in \mathbb{R}\), and the triangle inequality, we have
Hence, from Lemma 3 and (4.1), we obtain
uniformly over \( | v_j | \le C_j\), for any \( C_j <\infty\), \( j = 1, \dots, r\). Therefore, \( \widehat{F} _{T, H} (\varepsilon_1, \dots, \varepsilon_r) \) converges weakly to \( F (\varepsilon_1, \dots, \varepsilon_r)\), and the theorem is proved.
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Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 15-23 https://doi.org/10.4213/mzm13222.
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Garbaliauskienė, V., Šiaučiūnas, D. Joint Universality of Certain Dirichlet Series. Math Notes 111, 13–19 (2022). https://doi.org/10.1134/S0001434622010035
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DOI: https://doi.org/10.1134/S0001434622010035