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 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.
In the paper, we approximate analytic functions by generalized shifts $L(\lambda, \alpha, s+ig(\tau))$, $s=\sigma+it$, of the Lerch zeta-function, where $g$ is a certain increasing to $+\infty$ real function having a …
In the paper, we approximate analytic functions by generalized shifts $L(\lambda, \alpha, s+ig(\tau))$, $s=\sigma+it$, of the Lerch zeta-function, where $g$ is a certain increasing to $+\infty$ real function having a monotonic derivative. We prove that, for arbitrary parameters $\lambda$ and $\alpha$, there exists a closed set $\FF_{\lambda, \alpha}$ of analytic functions defined in the strip $1/2< \sigma<1$ which functions are approximated by the above shifts. If the set of logarithms $\log(m+\alpha)$, $m\in \NN_0$, is linearly independent over the field of rational numbers, then the set $\FF_{\lambda, \alpha}$ coincides with the set of all analytic functions in that strip.
In the paper, we approximate analytic functions by generalized shifts $L(\lambda, \alpha, s+ig(\tau))$, $s=\sigma+it$, of the Lerch zeta-function, where $g$ is a certain increasing to $+\infty$ real function having a …
In the paper, we approximate analytic functions by generalized shifts $L(\lambda, \alpha, s+ig(\tau))$, $s=\sigma+it$, of the Lerch zeta-function, where $g$ is a certain increasing to $+\infty$ real function having a monotonic derivative. We prove that, for arbitrary parameters $\lambda$ and $\alpha$, there exists a closed set $\FF_{\lambda, \alpha}$ of analytic functions defined in the strip $1/2< \sigma<1$ which functions are approximated by the above shifts. If the set of logarithms $\log(m+\alpha)$, $m\in \NN_0$, is linearly independent over the field of rational numbers, then the set $\FF_{\lambda, \alpha}$ coincides with the set of all analytic functions in that strip.
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.
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 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.
For $0$ < $\alpha,$ $\lambda \leq 1$, the Lerch zeta-function is defined by $L(s;\alpha, \lambda) := \sum_{n=0}^\infty e^{2\pi i\lambda n} (n+\alpha)^{-s}$, where $\sigma$ > $1$. In this paper, we prove …
For $0$ < $\alpha,$ $\lambda \leq 1$, the Lerch zeta-function is defined by $L(s;\alpha, \lambda) := \sum_{n=0}^\infty e^{2\pi i\lambda n} (n+\alpha)^{-s}$, where $\sigma$ > $1$. In this paper, we prove joint universality for Lerch zeta-functions with distinct $\lambda_1,\ldots,\lambda_m$ and transcendental $\alpha$.
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.
Let 0 < $\alpha$ < 1 be a transcendental real number and $\lambda_1,\ldots,\lambda_r$ be real numbers with $0 \le \lambda_j$ < 1. It is conjectured that a joint universality theorem …
Let 0 < $\alpha$ < 1 be a transcendental real number and $\lambda_1,\ldots,\lambda_r$ be real numbers with $0 \le \lambda_j$ < 1. It is conjectured that a joint universality theorem for a collection of Lerch zeta functions $\{L(\lambda_j,\alpha,s)\}$ will hold for every numbers $\lambda_j$'s which are different each other. In this paper we will prove that the joint universality theorem for the set $\{L(\lambda_j,\alpha,s)\}$ holds for almost all real numbers $\lambda_j$'s.
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.
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.
l Introduction.The function φ(x 9 a 9 s), defined for Hs > 1, x real, a ψ negative integer or zero, by the series o° 2nπix(1.1) φ (x,a s s) …
l Introduction.The function φ(x 9 a 9 s), defined for Hs > 1, x real, a ψ negative integer or zero, by the series o° 2nπix(1.1) φ (x,a s s) = Σ 1 7 y ' was investigated by Lipschitz [4;5], and Lerch [3].By use of the classic method of Riemann, φ{x, α, s) can be extended to the whole s-plane by means of the contour integralwhere the path C is a loop which begins at --00 , encircles the origin once in the positive direction, and returns to -00 .Since I(x 9 a, s) is an entire function of s, and we have d 3) φ(x,a,s)=Γ{l-s)l(x,a,s),this equation provides the analytic continuation of φ.For integer values of x, φ(x,a,s) is a meromorphic function (the Hurwitz zeta function) with only a simple pole at s -1.For nonintegral x it becomes an entire function of s.For 0 < x < 1, 0 < a < 1, we have the functional equation(1-4) φ{ x , a,l-s) first given by Lerch, whose proof follows the lines of the first Riemann proof of the functional equation for ζ(s) and uses Cauchy's theorem in connection with the contour integral (1.2).
One bright Sunday morning I went to church, And there I met a man named Lerch. We both did sing in jubilation, For he did show me a new equation. …
One bright Sunday morning I went to church, And there I met a man named Lerch. We both did sing in jubilation, For he did show me a new equation. Two simple derivations of the functional equation of <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts exp left-bracket 2 pi i n x right-bracket left-parenthesis n plus a right-parenthesis Superscript negative s"> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo movablelimits="false">∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:munderover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>exp</mml:mi> <mml:mo></mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum \limits _{n = 0}^\infty {\exp [2\pi inx]{{(n + a)}^{ - s}}}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> are given. The original proof is due to Lerch.
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.
The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the …
The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters and a step of arithmetic progression, there is a closed non-empty subset of the space of analytic functions defined in the critical strip such that its functions can be approximated by discrete shifts of the Lerch zeta function. The set of those shifts is infinite, and it has a positive density. For the proof, the weak convergence of probability measures in the space of analytic functions is 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
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.
После 1975 г. работы Воронина известно, что некоторые дзета и L-функции универсальны в том смысле, что их сдвигами приближается широкий класс аналитических функций. Рассматриваются два типа сдвигов: непрерывный и дискретный. …
После 1975 г. работы Воронина известно, что некоторые дзета и L-функции универсальны в том смысле, что их сдвигами приближается широкий класс аналитических функций. Рассматриваются два типа сдвигов: непрерывный и дискретный. В работе изучается универсальность дзета-функций Лерха L(λ,α,s), s = σ + it, которые в полуплоскости σ > 1 определяются рядами Дирихле с членами e 2πiλm (m + α) −s с фиксированными параметрами λ ∈ R и α, 0 < α ≤ 1, и мероморфно продолжаются на всю комплексную плоскость. Получены совместные дискретные теоремы универсальности для дзета-функций Лерха. Именно, набор аналитических функций f 1 (s),...,f r (s) одновременно приближаются сдвигами L(λ 1 ,α 1 ,s + ikh),...,L(λ r ,α r ,s + ikh), k = 0,1,2,..., где h > 0 - фиксированное число. При этом требуется линейная независимость над полем рациональных чисел множества{(log(m + α j ) : m ∈N 0 , j = 1,...,r), }. Доказательство теорем универсальности использует вероятностные предельные теоремы о слабой сходимости вероятностных мер в пространстве аналитических функций.
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.
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.
In the paper, a joint discrete universality theorem for periodic zeta-functions with multiplicative coefficients on the approximation of analytic functions by shifts involving the sequence f kg of imaginary parts …
In the paper, a joint discrete universality theorem for periodic zeta-functions with multiplicative coefficients on the approximation of analytic functions by shifts involving the sequence f kg of imaginary parts of nontrivial zeros of the Riemann zeta-function is obtained. For its proof, a weak form of the Montgomery pair correlation conjecture is used. The paper is a continuation of [A. Laurinčikas, M. Tekorė, Joint universality of periodic zeta-functions with multiplicative coefficients, Nonlinear Anal. Model. Control, 25(5):860–883, 2020] using nonlinear shifts for approximation of analytic functions.
We present the most general at this moment results on the discrete mixed joint value-distribution (Theorems 5 and 6) and the universality property (Theorems 3 and 4) for the class …
We present the most general at this moment results on the discrete mixed joint value-distribution (Theorems 5 and 6) and the universality property (Theorems 3 and 4) for the class of Matsumoto zeta-functions and periodic Hurwitz zeta-functions under certain linear independence condition on the relevant parameters, such as common differences of arithmetic progressions, prime numbers etc.
Let H(D) be the space of analytic functions on the strip ... In this paper, it is proved that there exists a closed non-empty set ...such that every collection of …
Let H(D) be the space of analytic functions on the strip ... In this paper, it is proved that there exists a closed non-empty set ...such that every collection of the functions ... is approximated by discrete shifts .., of Hurwitz zeta-functions with arbitrary parameters ...
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 We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurinčikas with respect to the length of these intervals. Moreover, we prove that …
Abstract We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurinčikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurinčikas in a problem session of a recent workshop on universality.
Abstract In this paper, we show an analog of hybrid universality theorem (see Ł. Pańkowski, Hybrid joint universality theorem for Dirichlet L-functions, Acta Arith. 141(1) (2010), pp. 59–72) for L-functions …
Abstract In this paper, we show an analog of hybrid universality theorem (see Ł. Pańkowski, Hybrid joint universality theorem for Dirichlet L-functions, Acta Arith. 141(1) (2010), pp. 59–72) for L-functions without Euler product like Lerch zeta-functions and periodic Hurwitz zeta-functions. More precisely, we prove that any analytic functions can be approximated by some L-functions shifted by iτ and, simultaneously, finitely many real numbers can be approximated by α1τ, …,α n τ, where α1, …,α n are real numbers linearly independent over ℚ. Keywords: hybrid universalityLerch zeta-functiondiophantine approximation2000 Mathematics Subject Classifications: Primary 11K6011M99 Acknowledgements This research was partially supported by the grant no. N N201 1482 33 from the Polish Ministry of Science and Higher Education.
The periodic zeta-function is defined by the ordinary Dirichlet series with periodic coefficients. In the paper, joint universality theorems on the approximation of a collection of analytic functions by nonlinear …
The periodic zeta-function is defined by the ordinary Dirichlet series with periodic coefficients. In the paper, joint universality theorems on the approximation of a collection of analytic functions by nonlinear shifts of periodic zeta-functions with multiplicative coefficients are obtained. These theorems do not use any independence hypotheses on the coefficients of zeta-functions.
In the paper, a theorem on approximation of a wide class of analytic functions by generalized shifts $\zeta_{u_T}(s+i\varphi(\tau))$ of an absolutely convergent Dirichlet series $\zeta_{u_T}(s)$ which in the mean is …
In the paper, a theorem on approximation of a wide class of analytic functions by generalized shifts $\zeta_{u_T}(s+i\varphi(\tau))$ of an absolutely convergent Dirichlet series $\zeta_{u_T}(s)$ which in the mean is close to the Riemann zeta-function is obtained. Here $\varphi(\tau)$ is a monotonically increasing differentiable function having a monotonic continuous derivative such that $\varphi(2\tau)\max\limits_{\tau\leqslant t\leqslant 2\tau} \frac{1}{\varphi'(t)} \ll \tau$ as $\tau\to\infty$, and $u_T\to\infty$ and $u_T\ll T^2$ as $T\to\infty$.
In the paper, we approximate analytic functions by generalized shifts $L(\lambda, \alpha, s+ig(\tau))$, $s=\sigma+it$, of the Lerch zeta-function, where $g$ is a certain increasing to $+\infty$ real function having a …
In the paper, we approximate analytic functions by generalized shifts $L(\lambda, \alpha, s+ig(\tau))$, $s=\sigma+it$, of the Lerch zeta-function, where $g$ is a certain increasing to $+\infty$ real function having a monotonic derivative. We prove that, for arbitrary parameters $\lambda$ and $\alpha$, there exists a closed set $\FF_{\lambda, \alpha}$ of analytic functions defined in the strip $1/2< \sigma<1$ which functions are approximated by the above shifts. If the set of logarithms $\log(m+\alpha)$, $m\in \NN_0$, is linearly independent over the field of rational numbers, then the set $\FF_{\lambda, \alpha}$ coincides with the set of all analytic functions in that strip.
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).