The joint universality theorem for Lerch zeta-functions L(λ l , α l , s) (1 ≤ l ≤ n) is proved, in the case when λ l s are rational …
The joint universality theorem for Lerch zeta-functions L(λ l , α l , s) (1 ≤ l ≤ n) is proved, in the case when λ l s are rational numbers and α l s are transcendental numbers. The case n = 1 was known before ([12]); the rationality of λ l s is used to establish the theorem for the “joint” case n ≥ 2. As a corollary, the joint functional independence for those functions is shown.
Abstract In this article, the universality in the Voronin sense for the Hurwitz zeta-function with periodic coefficients is proved. Keywords: Hurwitz zeta-functionProbability measureUniversalityWeak convergence Acknowledgement A. Laurinčikas is partially supported …
Abstract In this article, the universality in the Voronin sense for the Hurwitz zeta-function with periodic coefficients is proved. Keywords: Hurwitz zeta-functionProbability measureUniversalityWeak convergence Acknowledgement A. Laurinčikas is partially supported by a grant from Lithuanian Foundation of Studies and Science and by INTAS Grant no. 03-51-5070.
In this paper, an universality theorem in the Voronin sense for the periodic Hurwitz zeta-function is proved.
In this paper, an universality theorem in the Voronin sense for the periodic Hurwitz zeta-function is proved.
The periodic Hurwitz zeta function , s=σ+it, 0<α≤1, is defined, for σ>1, by and by analytic continuation elsewhere. Here {a m } is a periodic sequence of complex numbers. In …
The periodic Hurwitz zeta function , s=σ+it, 0<α≤1, is defined, for σ>1, by and by analytic continuation elsewhere. Here {a m } is a periodic sequence of complex numbers. In this paper, a discrete universality theorem for the function with a transcendental parameter α is proved. Roughly speaking, this means that every analytic function can be approximated uniformly on compact sets by shifts , where m is a non-negative integer and h is a fixed positive number such that is rational.
We obtain a joint universality theorem of Voronin type for a set of functions consisting of periodic zeta-functions and periodic Hurwitz zeta-functions with algebraically independent parameters.
We obtain a joint universality theorem of Voronin type for a set of functions consisting of periodic zeta-functions and periodic Hurwitz zeta-functions with algebraically independent parameters.
In this paper, the joint approximation of a given collection of analytic functions by a collection of shifts of zeta-functions with periodic coefficients is obtained. This is applied to prove …
In this paper, the joint approximation of a given collection of analytic functions by a collection of shifts of zeta-functions with periodic coefficients is obtained. This is applied to prove the functional independence for these zeta-functions.
The simultaneous universality of twisted automorphic L-functions, associated with a new form with respect to a congruence subgroup of SL(2,Z) and twisted by Dirichlet characters, is proved. Applications to the …
The simultaneous universality of twisted automorphic L-functions, associated with a new form with respect to a congruence subgroup of SL(2,Z) and twisted by Dirichlet characters, is proved. Applications to the functional independence and the zero density of linear combinations of those L-functions are given.
We prove a joint universality theorem for the Hurwitz zeta-functions with periodic coefficients.
We prove a joint universality theorem for the Hurwitz zeta-functions with periodic coefficients.
We apply an effective multidimensional Ω-result of Voronin in order to obtain effective universality-type theorems for the Riemann zeta-function.We further use this approach to study approximation properties of linear combinations …
We apply an effective multidimensional Ω-result of Voronin in order to obtain effective universality-type theorems for the Riemann zeta-function.We further use this approach to study approximation properties of linear combinations of derivatives of the zeta-function.
In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint …
In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint universality theorems on the approximation of a collection of analytic functions by discrete shifts of the above zeta-functions are obtained. For this, certain linear independence hypotheses are applied.
Let $0 \lt \gamma_1\leq \gamma_2 \leq\cdots$ be the imaginary parts of non-trivial zeros of the Riemann zeta-function $\zeta(s)$. Using the Montgomery conjecture (its weaker form) on the pair correlation of …
Let $0 \lt \gamma_1\leq \gamma_2 \leq\cdots$ be the imaginary parts of non-trivial zeros of the Riemann zeta-function $\zeta(s)$. Using the Montgomery conjecture (its weaker form) on the pair correlation of the sequence $\{\gamma_k\}$, we show that analyt
Algebraic number theory quadratic forms zeta and l-functions multiplicative number theory value distribution of arithmetic functions probabilistic theory of number systems and series miscellaneous. (Part contents).
Algebraic number theory quadratic forms zeta and l-functions multiplicative number theory value distribution of arithmetic functions probabilistic theory of number systems and series miscellaneous. (Part contents).
In this paper two weighted functional limit theorems for the function introduced by K. Matsumoto are proved.
In this paper two weighted functional limit theorems for the function introduced by K. Matsumoto are proved.
Abstract We prove a joint universality theorem in the Voronin sense for the periodic Hurwitz zeta-functions.
Abstract We prove a joint universality theorem in the Voronin sense for the periodic Hurwitz zeta-functions.
We prove the universality theorem for -functions of new parabolic forms. It concerns the uniform approximation of analytic functions by shifts of these -functions. This theorem together with the Shimura-Taniyama …
We prove the universality theorem for -functions of new parabolic forms. It concerns the uniform approximation of analytic functions by shifts of these -functions. This theorem together with the Shimura-Taniyama conjecture (now proved) yields the universality of -functions of non-singular elliptic curves over the field of rational numbers. The universality of -functions implies that they are functionally independent.
We prove a joint universality theorem for a collection of periodic Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers.
We prove a joint universality theorem for a collection of periodic Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers.
A result on the approximation of a fixed system of analytic functions by translations of Hurwitz zeta-functions with transcendental parameter is established. This is an analogue of Voronin's theorem on …
A result on the approximation of a fixed system of analytic functions by translations of Hurwitz zeta-functions with transcendental parameter is established. This is an analogue of Voronin's theorem on the joint universality of the Dirichlet -functions.Bibliography: 28 titles.
It is known that the Hurwitz zeta-function ζ(s, α) with transcendental or rational parameter α is universal in the sense that its shifts ζ(s + iτ, α), τ ∈ ℝ, …
It is known that the Hurwitz zeta-function ζ(s, α) with transcendental or rational parameter α is universal in the sense that its shifts ζ(s + iτ, α), τ ∈ ℝ, approximate with a given accuracy any analytic function uniformly on compact subsets of the strip D = {s ∈ ℂ : ½ < σ < 1}. Let H(D) denote the space of analytic functions on D equipped with the topology of uniform convergence on compacta. In the paper, the classes of functions F : H(D) → H(D) such that F(ζ(s, α)) is universal in the above sense are considered. For example, if F is continuous and, for each polynomial p = p(s), the set F -1 {p} is non-empty, then F(ζ(s, α)) with transcendental α is universal.
It is proved that a wide class of analytic functions can be approximated by shifts $\zeta (s+i\varphi (k))$, $k\geq k_0$, $k\in \mathbb {N}$, of the Riemann zeta-function. Here the function …
It is proved that a wide class of analytic functions can be approximated by shifts $\zeta (s+i\varphi (k))$, $k\geq k_0$, $k\in \mathbb {N}$, of the Riemann zeta-function. Here the function $\varphi (t)$ has a continuous nonvanishing derivative on $[k_0,\infty )$ satisfying the estimate $\varphi (2t) \max _{t\leq u \leq 2t} \left (\varphi '(u)\right )^{-1}\ll t$, and the sequence $\{a\varphi (k) : k\geq k_0\}$ with every real $a\neq 0$ is uniformly distributed modulo 1. Examples of $\varphi (t)$ are given.
For j=1,⋯,r, let Qj be a positive definite nj×nj matrix, and ζ(sj;Qj) denote the corresponding Epstein zeta-function. In this paper, assuming that nj⩾4 is even andx̲TQjx̲∈Z,x̲∈Zr∖{0̲}, a joint limit theorem …
For j=1,⋯,r, let Qj be a positive definite nj×nj matrix, and ζ(sj;Qj) denote the corresponding Epstein zeta-function. In this paper, assuming that nj⩾4 is even andx̲TQjx̲∈Z,x̲∈Zr∖{0̲}, a joint limit theorem of Bohr–Jessen type for the functions ζ(s1;Q1),⋯,ζ(sr;Qr), by using generalizing shifts ζ(σ1+iφ1(t);Q1),⋯,ζ(σr+iφr(t);Qr), is proved. Here, the functions φ1(t),⋯,φr(t) are increasing to +∞, with monotonic derivatives φj′(t) satisfying the asymptotic growth conditions: φj(t)≪tφj′(t), and φj′(t)=o(φj+1′(t)) as t→∞. An explicit form of the limit measure is given. This theorem extends and generalizes the previous result on the joint value-distribution of Epstein zeta-functions.
In the paper, we obtain a joint limit theorem on weak convergence for probability measure defined by discrete shifts of the Epstein and Hurwitz zeta-functions. The limit measure is explicitly …
In the paper, we obtain a joint limit theorem on weak convergence for probability measure defined by discrete shifts of the Epstein and Hurwitz zeta-functions. The limit measure is explicitly given. For the proof, some linear independence restriction is required. The proved theorem extends and continues Bohr–Jessen’s classical results on probabilistic characterization of value distribution for the Riemann zeta-function.
The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function ζ(s) (zeros different from s=−2m, m∈N) lie on the critical line σ=1/2. In this paper, …
The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function ζ(s) (zeros different from s=−2m, m∈N) lie on the critical line σ=1/2. In this paper, combining the universality property of ζ(s) with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts ζ(s+itτ) approximating the function ζ(s). Here, tτ denotes the Gram function, which is a continuous extension of the Gram points.
We consider the Beurling zeta function ζP(s), s=σ+it, of the system of generalized prime numbers P with generalized integers m satisfying the condition ∑m⩽x1=ax+O(xδ), a>0, 0⩽δ<1, and suppose that ζP(s) …
We consider the Beurling zeta function ζP(s), s=σ+it, of the system of generalized prime numbers P with generalized integers m satisfying the condition ∑m⩽x1=ax+O(xδ), a>0, 0⩽δ<1, and suppose that ζP(s) has a bounded mean square for σ>σP with some σP<1. Then, we prove that, for every h>0, there exists a closed non-empty set of analytic functions that are approximated by discrete shifts ζP(s+ilh). This set shifts has a positive density. For the proof, a weak convergence of probability measures in the space of analytic functions is applied.
In this study, the approximation of a pair of analytic functions defined on the strip {s=σ+it∈C:1/2<σ<1} by shifts (ζ(s+iτ),ζ(s+iτ,α)), τ∈R, of the Riemann and Hurwitz zeta-functions with transcendental α in …
In this study, the approximation of a pair of analytic functions defined on the strip {s=σ+it∈C:1/2<σ<1} by shifts (ζ(s+iτ),ζ(s+iτ,α)), τ∈R, of the Riemann and Hurwitz zeta-functions with transcendental α in the interval [T,T+H] with T27/82⩽H⩽T1/2 was considered. It was proven that the set of such shifts has a positive density. The main result was an extension of the Mishou theorem proved for the interval [0,T], and the first theorem on the joint mixed universality in short intervals. For proof, the probability approach was applied.
By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function ζ(s) by shifts ζ(s+iτ). In this paper, …
By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function ζ(s) by shifts ζ(s+iτ). In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating ζ(s) with all but at most countably many accuracies ε>0. Also, the analogue of an equivalent in terms of positive density in short intervals is discussed.
The Hurwitz zeta-function ζ(s,α), s=σ+it, with parameter 0<α⩽1 is a generalization of the Riemann zeta-function ζ(s) (ζ(s,1)=ζ(s)) and was introduced at the end of the 19th century. The function ζ(s,α) …
The Hurwitz zeta-function ζ(s,α), s=σ+it, with parameter 0<α⩽1 is a generalization of the Riemann zeta-function ζ(s) (ζ(s,1)=ζ(s)) and was introduced at the end of the 19th century. The function ζ(s,α) plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function ζ(s,α) is the main example of zeta-functions without Euler’s product (except for the cases α=1, α=1/2), and its value distribution is governed by arithmetical properties of α. For the majority of zeta-functions, ζ(s,α) for some α is universal, i.e., its shifts ζ(s+iτ,α), τ∈R, approximate every analytic function defined in the strip {s:1/2<σ<1}. For needs of effectivization of the universality property for ζ(s,α), the interval for τ must be as short as possible, and this can be achieved by using the mean square estimate for ζ(σ+it,α) in short intervals. In this paper, we obtain the bound O(H) for that mean square over the interval [T−H,T+H], with T27/82⩽H⩽Tσ and 1/2<σ⩽7/12. This is the first result on the mean square for ζ(s,α) in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for ζ(s,α) and other zeta-functions in short intervals.
In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on C2 defined by means of the Epstein ζ(s;Q) and Hurwitz ζ(s,α) …
In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on C2 defined by means of the Epstein ζ(s;Q) and Hurwitz ζ(s,α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter α are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function.
In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z(s), s=σ+it, with fixed 1/2<σ<1, of the …
In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z(s), s=σ+it, with fixed 1/2<σ<1, of the square |ζ(1/2+it)|2 of the Riemann zeta-function. We consider probability measures defined by means of Z(σ+iφ(t)), where φ(t), t⩾t0>0, is an increasing to +∞ differentiable function with monotonically decreasing derivative φ′(t) satisfying a certain normalizing estimate related to the mean square of the function Z(σ+iφ(t)). This allows us to extend the distribution laws for Z(s).
The Lerch zeta-function $L(\lambda, \alpha,s)$, $s=\sigma+it$, depends on two real parameters $\lambda$ and $0<\alpha\leqslant 1$, and, for $\sigma>1$, is defined by the Dirichlet series $\sum_{m=0}^\infty \ee^{2\pi i\lambda m} (m+\alpha)^{-s}$, and …
The Lerch zeta-function $L(\lambda, \alpha,s)$, $s=\sigma+it$, depends on two real parameters $\lambda$ and $0<\alpha\leqslant 1$, and, for $\sigma>1$, is defined by the Dirichlet series $\sum_{m=0}^\infty \ee^{2\pi i\lambda m} (m+\alpha)^{-s}$, and by analytic continuation elsewhere. In the paper, we consider the joint approximation of collections of analytic functions by discrete shifts $(L(\lambda_1, \alpha_1, s+ikh_1), \dots, L(\lambda_r, \alpha_r, s+ikh_r))$, $k=0, 1, \dots$, with arbitrary $\lambda_j$, $0<\alpha_j\leqslant 1$ and $h_j>0$, $j=1, \dots, r$. We prove that there exists a non-empty closed set of analytic functions on the critical strip $1/2<\sigma<1$ which is approximated by the above shifts. It is proved that the set of shifts approximating a given collection of analytic functions has a positive lower density. The case of positive density also is discussed. A generalization for some compositions is given.
Let P be the set of generalized prime numbers, and ζP(s), s=σ+it, denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by …
Let P be the set of generalized prime numbers, and ζP(s), s=σ+it, denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζP(s+iτ), τ∈R. We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set {logp:p∈P}, and the existence of a bounded mean square for ζP(s). Under the above hypotheses, we obtain the universality of the function ζP(s). This means that the set of shifts ζP(s+iτ) approximating a given analytic function defined on a certain strip σ^<σ<1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied.
In this paper, the approximation of analytic functions by shifts ζP(s+iτ) of Beurling zeta-functions ζP(s) of certain systems P of generalized prime numbers is discussed. It is required that the …
In this paper, the approximation of analytic functions by shifts ζP(s+iτ) of Beurling zeta-functions ζP(s) of certain systems P of generalized prime numbers is discussed. It is required that the system of generalized integers NP generated by P satisfies ∑m⩽x,m∈N1=ax+O(xδ), a>0, 0⩽δ<1, and the function ζP(s) in some strip lying in σ^<σ<1, σ^>δ, which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces.
Abstract In this article, we consider the asymptotic behaviour of the modified Mellin transform <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">Z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathcal{Z}}\left(s) , <m:math …
Abstract In this article, we consider the asymptotic behaviour of the modified Mellin transform <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">Z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathcal{Z}}\left(s) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> <m:mo>=</m:mo> <m:mi>σ</m:mi> <m:mo>+</m:mo> <m:mi>i</m:mi> <m:mi>t</m:mi> </m:math> s=\sigma +it , of the Riemann zeta-function using weak convergence of probability measures in the space of analytic functions defined by means of shifts <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">Z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>+</m:mo> <m:mi>i</m:mi> <m:mi>φ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>τ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathcal{Z}}\left(s+i\varphi \left(\tau )) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>φ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>τ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \varphi \left(\tau ) is a real increasing to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>+</m:mo> <m:mi>∞</m:mi> </m:math> +\infty differentiable function with monotonically decreasing derivative satisfying a certain estimate connected to the second moment of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">Z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathcal{Z}}\left(s) . We prove in this case that the limit measure is concentrated at the point <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>≡</m:mo> <m:mn>0</m:mn> </m:math> {g}_{0}\left(s)\equiv 0 . This result is applied to the approximation of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {g}_{0}\left(s) by shifts <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">Z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>+</m:mo> <m:mi>i</m:mi> <m:mi>φ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>τ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathcal{Z}}\left(s+i\varphi \left(\tau )) .
In the paper, we prove that the set of discrete shifts of the Riemann zeta-function approximating analytic nonvanishing functions f1(s),...,fr(s) defined on has a positive density in the interval [N,N …
In the paper, we prove that the set of discrete shifts of the Riemann zeta-function approximating analytic nonvanishing functions f1(s),...,fr(s) defined on has a positive density in the interval [N,N + M] with with real algebraic numbers a1,...,ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series.
AbstractIn the paper, we prove a general universality theorem for the Riemann zeta-function ζ (s) on approximation of a class of analytic functions by generalized shifts ζ (s + ia(τ)). …
AbstractIn the paper, we prove a general universality theorem for the Riemann zeta-function ζ (s) on approximation of a class of analytic functions by generalized shifts ζ (s + ia(τ)). Here a(τ) is a real-valued continuous increasing to +∞ function, uniformly distributed modulo 1 and such that |ζ(σ + ia(τ) + it)|2, for σ > 1/2, has a traditional mean estimate for every t ∈ ℝ. For example, the function tv(τ), v > 0, where t(τ) is the Gram function, satisfies the hypotheses of the proved theorem. For the proof, the method of weak convergence of probabilistic measures in the space of analytic functions is developed.Mathematics Subject Classification (2020): 11M06Key words: Approximation of analytic functionsRiemann zeta-functionuniversalityweak convergence of probability measures
In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic …
In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical zero in the strip {s∈C:1/2<σ<1}.
In the paper, we consider the approximation of analytic functions by shifts from the wide class S˜ of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, …
In the paper, we consider the approximation of analytic functions by shifts from the wide class S˜ of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint discrete universality theorem for the function L(s)∈S˜. Using the linear independence over Q of the multiset (hjlogp:p∈P),j=1,…,r;2π for positive hj, we obtain that there are many infinite shifts L(s+ikh1),…,L(s+ikhr), k=0,1,…, approximating every collection f1(s),…,fr(s) of analytic non-vanishing functions defined in the strip {s∈C:σL<σ<1}, where σL is a degree of the function L(s). For the proof, the probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied.
This paper is devoted to the approximation of a certain class of analytic functions by shifts Z(s+iτ), τ∈R, of the modified Mellin transform Z(s) of the square of the Riemann …
This paper is devoted to the approximation of a certain class of analytic functions by shifts Z(s+iτ), τ∈R, of the modified Mellin transform Z(s) of the square of the Riemann zeta-function ζ(1/2+it). More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z(s+iτ), which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure.
In the paper, the approximation of analytic functions on compact sets of the strip {s=σ+it∈C∣1/2<σ<1} by shifts F(ζ(s+iu1(τ)),…,ζ(s+iur(τ))), where ζ(s) is the Riemann zeta-function, u1,…,ur are certain differentiable increasing functions, …
In the paper, the approximation of analytic functions on compact sets of the strip {s=σ+it∈C∣1/2<σ<1} by shifts F(ζ(s+iu1(τ)),…,ζ(s+iur(τ))), where ζ(s) is the Riemann zeta-function, u1,…,ur are certain differentiable increasing functions, and F is a certain continuous operator in the space of analytic functions, is considered. It is obtained that the set of the above shifts in the interval [T,T+H] with H=o(T), T→∞, has a positive lower density. Additionally, the positivity of a density with a certain exceptional condition is discussed. Examples of considered operators F are given.
In the paper, it is obtained that there are infinite discrete shifts Ξ(s+ikh), h>0, k∈N0 of the Mellin transform Ξ(s) of the square of the Riemann zeta-function, approximating a certain …
In the paper, it is obtained that there are infinite discrete shifts Ξ(s+ikh), h>0, k∈N0 of the Mellin transform Ξ(s) of the square of the Riemann zeta-function, approximating a certain class of analytic functions. For the proof, a probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied.
In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts (ζ(s+itkα1),…,ζ(s+itkαr)) of the Riemann zeta function. Here, {tk:k∈N} is …
In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts (ζ(s+itkα1),…,ζ(s+itkαr)) of the Riemann zeta function. Here, {tk:k∈N} is the sequence of Gram numbers, and α1,…,αr are different positive numbers not exceeding 1. It is proved that the above set of shifts in the interval [N,N+M], here M=o(N) as N→∞, has a positive lower density. For the proof, a joint limit theorem in short intervals for weakly convergent probability measures is applied.
In the paper, we consider a collection of absolutely convergent Dirichlet series which in the mean are close to periodic and periodic Hurwitz zeta-functions, and prove that the shifts of …
In the paper, we consider a collection of absolutely convergent Dirichlet series which in the mean are close to periodic and periodic Hurwitz zeta-functions, and prove that the shifts of mentioned Dirichlet series approximate simultaneously a wide class of analytic functions.
Suppose that Q is a positive defined n×n matrix, and Q[x̲]=x̲TQx̲ with x̲∈Zn. The Epstein zeta-function ζ(s;Q), s=σ+it, is defined, for σ>n2, by the series ζ(s;Q)=∑x̲∈Zn∖{0̲}(Q[x̲])−s, and it has a …
Suppose that Q is a positive defined n×n matrix, and Q[x̲]=x̲TQx̲ with x̲∈Zn. The Epstein zeta-function ζ(s;Q), s=σ+it, is defined, for σ>n2, by the series ζ(s;Q)=∑x̲∈Zn∖{0̲}(Q[x̲])−s, and it has a meromorphic continuation to the whole complex plane. Let n⩾4 be even, while φ(t) is an increasing differentiable function with a continuous monotonic bounded derivative φ′(t) such that φ(2t)(φ′(t))−1≪t, and the sequence {aφ(k)} is uniformly distributed modulo 1. In the paper, it is obtained that 1N#N⩽k⩽2N:ζ(σ+iφ(k);Q)∈A, A∈B(C), for σ>n−12, converges weakly to an explicitly given probability measure on (C,B(C)) as N→∞.
In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed …
In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand side of the critical strip, which is approximated by the above tuples of shifts. Further, a generalization for some compositions of tuples of Lerch zeta-functions is given.
The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class S is a complemented Selberg …
The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class S is a complemented Selberg class by an arithmetic hypothesis related to the distribution of prime numbers. In this paper, a joint universality theorem for the functions L from the class S on the approximation of a collection of analytic functions by shifts L(s+ia1τ),…,L(s+iarτ), where a1,…,ar are real algebraic numbers linearly independent over the field of rational numbers, is obtained. It is proved that the set of the above approximating shifts is infinite, its lower density and, with some exception, density are positive. For the proof, a probabilistic method based on weak convergence of probability measures in the space of analytic functions is applied together with the Backer theorem on linear forms of logarithms and the Mergelyan theorem on approximation of analytic functions by polynomials.
Let θ(t) denote the increment of the argument of the product π−s/2Γ(s/2) along the segment connecting the points s=1/2 and s=1/2+it, and tn denote the solution of the equation θ(t)=(n−1)π, …
Let θ(t) denote the increment of the argument of the product π−s/2Γ(s/2) along the segment connecting the points s=1/2 and s=1/2+it, and tn denote the solution of the equation θ(t)=(n−1)π, n=0,1,…. The numbers tn are called the Gram points. In this paper, we consider the approximation of a collection of analytic functions by shifts in the Riemann zeta-function (ζ(s+itkα1),…,ζ(s+itkαr)), k=0,1,…, where α1,…,αr are different positive numbers not exceeding 1. We prove that the set of such shifts approximating a given collection of analytic functions has a positive lower density. For the proof, a discrete limit theorem on weak convergence of probability measures in the space of analytic functions is applied.
In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of …
In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of a wide class of analytic functions.
Let tτ be a solution to the equation θ(t)=(τ−1)π, τ>0, where θ(t) is the increment of the argument of the function π−s/2Γ(s/2) along the segment connecting points s=1/2 and s=1/2+it. …
Let tτ be a solution to the equation θ(t)=(τ−1)π, τ>0, where θ(t) is the increment of the argument of the function π−s/2Γ(s/2) along the segment connecting points s=1/2 and s=1/2+it. tτ is called the Gram function. In the paper, we consider the approximation of collections of analytic functions by shifts of the Riemann zeta-function (ζ(s+itτα1),…,ζ(s+itταr)), where α1,…,αr are different positive numbers, in the interval [T,T+H] with H=o(T), T→∞, and obtain the positivity of the density of the set of such shifts. Moreover, a similar result is obtained for shifts of a certain absolutely convergent Dirichlet series connected to ζ(s). Finally, an example of the approximation of analytic functions by a composition of the above shifts is given.
Let Q be a positive defined n×n matrix and Q[x̲]=x̲TQx̲. The Epstein zeta-function ζ(s;Q), s=σ+it, is defined, for σ>n2, by the series ζ(s;Q)=∑x̲∈Zn\{0̲}(Q[x̲])−s, and is meromorphically continued on the whole …
Let Q be a positive defined n×n matrix and Q[x̲]=x̲TQx̲. The Epstein zeta-function ζ(s;Q), s=σ+it, is defined, for σ>n2, by the series ζ(s;Q)=∑x̲∈Zn\{0̲}(Q[x̲])−s, and is meromorphically continued on the whole complex plane. Suppose that n⩾4 is even and φ(t) is a differentiable function with a monotonic derivative. In the paper, it is proved that 1Tmeast∈[0,T]:ζ(σ+iφ(t);Q)∈A, A∈B(C), converges weakly to an explicitly given probability measure on (C,B(C)) as T→∞.
In the paper, the simultaneous approximation of a tuple of analytic functions in the strip {s=σ+it∈C:1/2<σ<1} by shifts (ζ(s+iφ1(τ)),…,ζ(s+iφr(τ))) of the Riemann zeta-function ζ(s) with a certain class of continuously …
In the paper, the simultaneous approximation of a tuple of analytic functions in the strip {s=σ+it∈C:1/2<σ<1} by shifts (ζ(s+iφ1(τ)),…,ζ(s+iφr(τ))) of the Riemann zeta-function ζ(s) with a certain class of continuously differentiable increasing functions φ1,…,φr is considered. This class of functions φ1,…,φr is characterized by the growth of their derivatives. It is proved that the set of mentioned shifts in the interval [T,T+H] with H=o(T) has a positive lower density. The precise expression for H is described by the functions (φj(τ))1/3(logφj(τ))26/15 and derivatives φj′(τ). The density problem is also discussed. An example of the approximation by a composition F(ζ(s+iφ1(τ)),…,ζ(s+iφr(τ))) with a certain continuous operator F in the space of analytic functions is given.
In the paper, an universality theorem of discrete type on the approximation of analytic functions by shifts of a special absolutely convergent Dirichlet series is obtained. These series is close …
In the paper, an universality theorem of discrete type on the approximation of analytic functions by shifts of a special absolutely convergent Dirichlet series is obtained. These series is close in a certain sense to the periodic zeta-function and depends on a parameter.
Abstract In the paper, collections of analytic functions are simultaneously approximated by collections of shifts of Dirichlet L -functions ( L ( s + i γ 1 ( τ ), …
Abstract In the paper, collections of analytic functions are simultaneously approximated by collections of shifts of Dirichlet L -functions ( L ( s + i γ 1 ( τ ), χ 1 ),…, L ( s + i γ r ( τ ), χ r )), with arbitrary Dirichlet characters χ 1 ,…, χ r . The differentiable functions γ 1 ( τ ), …, γ r ( τ ) and their derivatives satisfy certain growth conditions. The obtained results extend those of [PAŃKOWSKI, Ł.: Joint universality for dependent L-functions , Ramanujan J. 45 (2018), 181–195].
We consider a certain Dirichlet series associated with the zeta function of a normalized Hecke cusp form. It is absolutely convergent on the right of the critical strip. We obtain …
We consider a certain Dirichlet series associated with the zeta function of a normalized Hecke cusp form. It is absolutely convergent on the right of the critical strip. We obtain universality theorems on the approximation of a wide class of analytic functions by shifts of this series. Bibliography: 9 titles.
In the paper, a theorem on the approximation of collections of a wide class of analytic functions by collections of shifts of zeta-functions with periodic co-efficients involving imaginary parts of …
In the paper, a theorem on the approximation of collections of a wide class of analytic functions by collections of shifts of zeta-functions with periodic co-efficients involving imaginary parts of non-trivial zeros of the Riemann zeta-function is obtained. For this, a version of the Montgomery pair correlation conjecture is required.
Известно, что дзета-функция Гурвица $\zeta (s,\alpha )$ с трансцендентным или рациональным параметром $\alpha $ обладает свойством дискретной универсальности, т.е. сдвиги $\zeta (s+ikh,\alpha )$, $k\in \mathbb N_0$, $h>0$, аппроксимируют широкий класс …
Известно, что дзета-функция Гурвица $\zeta (s,\alpha )$ с трансцендентным или рациональным параметром $\alpha $ обладает свойством дискретной универсальности, т.е. сдвиги $\zeta (s+ikh,\alpha )$, $k\in \mathbb N_0$, $h>0$, аппроксимируют широкий класс аналитических функций. Случай алгебраического иррационального $\alpha $ - сложная открытая проблема. В работе получено некоторое продвижение в этой проблеме. Доказано, что существует непустое замкнутое множество аналитических функций $F_{\alpha ,h}$, аппроксимируемых указанными выше сдвигами. Также обсуждается случай некоторых композиций $\Phi (\zeta (s,\alpha ))$.
Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.
Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.
This paper considers the question of approximating analytic functions by translations of the Riemann zeta-function. Bibliography: 6 items.
This paper considers the question of approximating analytic functions by translations of the Riemann zeta-function. Bibliography: 6 items.
The joint universality theorem for Lerch zeta-functions L(λ l , α l , s) (1 ≤ l ≤ n) is proved, in the case when λ l s are rational …
The joint universality theorem for Lerch zeta-functions L(λ l , α l , s) (1 ≤ l ≤ n) is proved, in the case when λ l s are rational numbers and α l s are transcendental numbers. The case n = 1 was known before ([12]); the rationality of λ l s is used to establish the theorem for the “joint” case n ≥ 2. As a corollary, the joint functional independence for those functions is shown.
We prove that, for arbitrary Dirichlet L-functions $$L(s;\chi _1),\ldots ,L(s;\chi _n)$$ (including the case when $$\chi _j$$ is equivalent to $$\chi _k$$ for $$j\ne k$$ ), suitable shifts of type …
We prove that, for arbitrary Dirichlet L-functions $$L(s;\chi _1),\ldots ,L(s;\chi _n)$$ (including the case when $$\chi _j$$ is equivalent to $$\chi _k$$ for $$j\ne k$$ ), suitable shifts of type $$L(s+i\alpha _jt^{a_j}\log ^{b_j}t;\chi _j)$$ can simultaneously approximate any given set of analytic functions on a simply connected compact subset of the right open half of the critical strip, provided the pairs $$(a_j,b_j)$$ are distinct and satisfy certain conditions. Moreover, we consider a discrete analogue of this problem where t runs over the set of positive integers.
The periodic Hurwitz zeta function , s=σ+it, 0<α≤1, is defined, for σ>1, by and by analytic continuation elsewhere. Here {a m } is a periodic sequence of complex numbers. In …
The periodic Hurwitz zeta function , s=σ+it, 0<α≤1, is defined, for σ>1, by and by analytic continuation elsewhere. Here {a m } is a periodic sequence of complex numbers. In this paper, a discrete universality theorem for the function with a transcendental parameter α is proved. Roughly speaking, this means that every analytic function can be approximated uniformly on compact sets by shifts , where m is a non-negative integer and h is a fixed positive number such that is rational.
Abstract In this article, the universality in the Voronin sense for the Hurwitz zeta-function with periodic coefficients is proved. Keywords: Hurwitz zeta-functionProbability measureUniversalityWeak convergence Acknowledgement A. Laurinčikas is partially supported …
Abstract In this article, the universality in the Voronin sense for the Hurwitz zeta-function with periodic coefficients is proved. Keywords: Hurwitz zeta-functionProbability measureUniversalityWeak convergence Acknowledgement A. Laurinčikas is partially supported by a grant from Lithuanian Foundation of Studies and Science and by INTAS Grant no. 03-51-5070.
In this paper, an universality theorem in the Voronin sense for the periodic Hurwitz zeta-function is proved.
In this paper, an universality theorem in the Voronin sense for the periodic Hurwitz zeta-function is proved.
in KOPENHAGEN.(Erste Mi~teilung.Das Verhalten der Funl~ion in der Halbebene 0> I .) InhaltsUbersicht.Einleitung.Erster Teil.Das Verhalten der Zetafunktion auf einer vertikalen Geraden a=eo(> ~).w i. Excurs iiber die Methode.w w Zweiter …
in KOPENHAGEN.(Erste Mi~teilung.Das Verhalten der Funl~ion in der Halbebene 0> I .) InhaltsUbersicht.Einleitung.Erster Teil.Das Verhalten der Zetafunktion auf einer vertikalen Geraden a=eo(> ~).w i. Excurs iiber die Methode.w w Zweiter w w w w w 2. Versch~irfung der Hilfsmittel.3. Erster Hauptsatz.Wahrscheinlichkeitsverteilungen auf vertikalen Geraden.Teil.Das Verhalten der Zetafunktion in einem vertik.Streifen (~<) a~<(~, a~. 4. Analytische Vorbereitungen.5. Erste Anwendung tier Hilfss~itze auf die Zetafunktion.6. Zweiter ttauptsatz.Wahrscheinlichkeitsverteilungen in vertikalen Streifen.7. Anwendung des ersten Hauptsatzes zum Beweis
0 f(x)x s 1 dx with s = + it denote the Mellin transform of f(x). Mellin transforms play a fundamental role in Analytic Number Theory. They can be viewed, …
0 f(x)x s 1 dx with s = + it denote the Mellin transform of f(x). Mellin transforms play a fundamental role in Analytic Number Theory. They can be viewed, by a change of variable, as special cases of Fourier transforms, and their properties can be deduced from the general theory of Fourier transforms. For an extensive account, we refer the reader to E. C. Titchmarsh (25). One of the basic properties of Mellin transforms is the inversion formula 1 2 {f(x + 0) +f(x 0)} = 1 2i ( ) F (s)x s ds = 1 2i lim T!1 +iT
Abstract It is well known that Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers are universal in the sense that their shifts approximate simultaneously any collection …
Abstract It is well known that Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers are universal in the sense that their shifts approximate simultaneously any collection of analytic functions. In this paper we introduce some classes of universal composite functions of a collection of Hurwitz zeta-functions.
Some results and conjectures on $Z_2(s) = \int_1^\infty |\zeta(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are presented. Consequences of these conjectures regarding the eighth moment of $|\zeta(1/2+it)$ and the error term in …
Some results and conjectures on $Z_2(s) = \int_1^\infty |\zeta(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are presented. Consequences of these conjectures regarding the eighth moment of $|\zeta(1/2+it)$ and the error term in the fourth moment of $|\zeta(1/2+it)$ are discussed.
We obtain a joint universality theorem of Voronin type for a set of functions consisting of periodic zeta-functions and periodic Hurwitz zeta-functions with algebraically independent parameters.
We obtain a joint universality theorem of Voronin type for a set of functions consisting of periodic zeta-functions and periodic Hurwitz zeta-functions with algebraically independent parameters.
Article Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion. was published on January 1, 1914 in the journal Journal für die reine und angewandte Mathematik (volume 1914, …
Article Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion. was published on January 1, 1914 in the journal Journal für die reine und angewandte Mathematik (volume 1914, issue 144).
We complete the proof that every elliptic curve over the rational numbers is modular.
We complete the proof that every elliptic curve over the rational numbers is modular.
We prove results on the transcendence degree of a field generated by numbers connected with the modular function . In particular, we show that and are algebraically independent and we …
We prove results on the transcendence degree of a field generated by numbers connected with the modular function . In particular, we show that and are algebraically independent and we prove Bertrand's conjecture on algebraic independence over of the values at algebraic points of a modular function and its derivatives.
The simultaneous universality of twisted automorphic L-functions, associated with a new form with respect to a congruence subgroup of SL(2,Z) and twisted by Dirichlet characters, is proved. Applications to the …
The simultaneous universality of twisted automorphic L-functions, associated with a new form with respect to a congruence subgroup of SL(2,Z) and twisted by Dirichlet characters, is proved. Applications to the functional independence and the zero density of linear combinations of those L-functions are given.