Search

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 + i τ) . 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. [ABSTRACT FROM AUTHOR]

In this article, we consider the asymptotic behaviour of the modified Mellin transform Z(s){\mathcal{Z}}\left(s), s=σ+its=\sigma +it, of the Riemann zeta-function using weak convergence of probability measures in the space of analytic functions defined by means of shifts Z(s+iφ(τ)){\mathcal{Z}}\left(s+i\varphi \left(\tau )), where φ(τ)\varphi \left(\tau ) is a real increasing to +∞+\infty differentiable function with monotonically decreasing derivative satisfying a certain estimate connected to the second moment of Z(s){\mathcal{Z}}\left(s). We prove in this case that the limit measure is concentrated at the point g0(s)≡0{g}_{0}\left(s)\equiv 0. This result is applied to the approximation of g0(s){g}_{0}\left(s) by shifts Z(s+iφ(τ)){\mathcal{Z}}\left(s+i\varphi \left(\tau )).

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.

In the paper, we prove that the set of discrete shifts of the Riemann zeta-function (ζ(s + 2πia1k), . . ., ζ(s + 2πiark)), k ∈ N, approximating analytic non-vanishing functions f1(s), . . ., fr(s) defined on {s ∈ C : 1/2 < Res < 1} has a positive density in the interval [N,N + M] with M = o(N), N → ∞, with real algebraic numbers a1, . . ., ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series. [ABSTRACT FROM AUTHOR]

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.

This paper is devoted to the approximation of a certain class of analytic functions by shifts Z (s + i τ) , τ ∈ R , of the modified Mellin transform Z (s) of the square of the Riemann zeta-function ζ (1 / 2 + i t) . More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z (s + i τ) , 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. [ABSTRACT FROM AUTHOR]

In the paper, it is obtained that there are infinite discrete shifts Ξ (s + i k h) , h > 0 , k ∈ N 0 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. [ABSTRACT FROM AUTHOR]

In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts (ζ (s + i t k α 1) , ... , ζ (s + i t k α r)) of the Riemann zeta function. Here, { t k : k ∈ N } is the sequence of Gram numbers, and α 1 , ... , α r 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. [ABSTRACT FROM AUTHOR]

Let t τ 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 / 2 Γ (s / 2) along the segment connecting points s = 1 / 2 and s = 1 / 2 + i t . t τ 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 + i t τ α 1) , ... , ζ (s + i t τ α r)) , where α 1 , ... , α r 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. [ABSTRACT FROM AUTHOR]

In the paper, a theorem on approximation of a wide class of analytic functions by generalized shifts uT (s + if(t )) of an absolutely convergent Dirichlet series uT (s) which in the mean is close to the Riemann zetafunction is obtained. Here f(t) is a monotonically increasing differentiable function having a monotonic continuous derivative such that f(2t) max tt2t 1 f'(t) « t as t 8, and uT 8 and uT « T2 as T 8. [ABSTRACT FROM AUTHOR]

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, 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 the paper, the simultaneous approximation of a tuple of analytic functions in the strip { s = σ + i t ∈ C : 1 / 2 < σ < 1 } by shifts (ζ (s + i φ 1 (τ)) , ... , ζ (s + i φ r (τ))) of the Riemann zeta-function ζ (s) with a certain class of continuously differentiable increasing functions φ 1 , ... , φ r is considered. This class of functions φ 1 , ... , φ r 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 (φ j (τ)) 1 / 3 (log φ j (τ)) 26 / 15 and derivatives φ j ′ (τ) . The density problem is also discussed. An example of the approximation by a composition F (ζ (s + i φ 1 (τ)) , ... , ζ (s + i φ r (τ))) with a certain continuous operator F in the space of analytic functions is given. [ABSTRACT FROM AUTHOR]

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 2007, H. Mishou proved the universality theorem on the joint approximation of a pair of analytic functions by the shifts of the Riemann zeta-function and the Hurwitz zeta-function with transcendental parameter . In this paper, we obtain a similar theorem on approximation by the shifts , , , where and are absolutely convergent Dirichlet series, and, as , they tend in mean to and respectively. [ABSTRACT FROM AUTHOR]

By the Mishou theorem (Mishou in Lith Math J 47:32–47, 2007), the set of shifts (ζ (s + i τ) , ζ (s + i τ , α)) , τ ∈ R , of the Riemann zeta-function and Hurwitz zeta-function with transcendental parameter simultaneously approximating a pair of analytic functions has a positive lower density. In the paper, two absolutely convergent Dirichlet series close to the Riemann and Hurwitz zeta-functions are constructed, and it is proved that the set of their shifts approximating a pair of analytic functions has a positive density. For the proof, probabilistic arguments are applied. [ABSTRACT FROM AUTHOR]

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.

In the paper, an absolutely convergent Dirichlet series whose shifts approximate a wide class of analytic functions is constructed. This series is close in the mean to the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

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.

It is well known that the Riemann zeta-function is universal in the Voronin sense, i.e., its shifts ζ(s + iτ), τ ∈ R, approximate a wide class of analytic functions. The universality of ζ(s) is called discrete if τ take values from a certain discrete set. In the paper, we obtain a weighted discrete universality theorem for ζ(s) when τ takes values from the arithmetic progression {kh : k ∈N} with arbitrary fixed h > 0. For this, two types of h are considered.

Let 0 < γ1< γ2< ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + iγkh, α), h > 0, with parameter α such that the set {log(m + α): m ∈ ℕ 0 } is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied. [ABSTRACT FROM AUTHOR]

By the Voronin theorem, the set of shifts of the Riemann zeta-function ζ (s + i τ) , s = σ + i t , τ ∈ R , that approximate any given non-vanishing analytic function defined on { s ∈ C : 1 2 < σ < 1 } has a positive lower density. In the paper, it is proved that the same property of the above shifts remains valid in the intervals [ T , T + H ] with T 1 / 3 (log ⁡ T) 26 / 15 ⩽ H ⩽ T. [ABSTRACT FROM AUTHOR]

In the paper, a Dirichlet series ζuN(s) whose shifts ζuN(s+ikh), k=0,1,⋯, h>0, approximate analytic non-vanishing functions defined on the right-hand side of the critical strip is considered. This series is closely connected to the Riemann zeta-function. The sequence uN→∞ and uN≪N2 as N→∞.

We prove that every collection of analytic functions (f1(s),... , fr(s)) defined on the right-hand side of the critical strip can be simultaneously approximated by shifts of Hurwitz zeta-functions (ζ(s + iγκh, α1), ... , ζ(s + iγκh, αr)), h > 0, where 0 < γ1 ≤ γ2 ≤ ... are the imaginary parts of nontrivial zeros of the Riemann zeta-function ζ(s). We use the weak form of the Montgomery pair correlation conjecture and the linear independence over ℚ of the set {log(m + αj) : m ∈ ℕ0, j = 1, ... , r}. [ABSTRACT FROM AUTHOR]

We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts ζ(s + iγk), k ∈ ℕ, of the Riemann zeta-function, where γk are imaginary parts of nontrivial zeros of ζ(s). [ABSTRACT FROM AUTHOR]

In the paper, a two-dimentional discrete limit theorem in the sense of weak convergence of probability measures in the space of analytic functions for Mellin transforms of the Riemann zeta-function on the critical line is obtained.

In the paper two-dimensional limit theorem for the modified Mellin transform of the Riemann zeta-function is obtained.

In the paper, a multiplicative version of the distribution of the vector (|ζ(s)|, ζ(s)), where ζ(s) is the Riemann zeta-function, is considered and probabilistic limit theorems are obtained. The dependence between |ζ(s)| and ζ(s) in terms of m-characteristic transforms is also discussed. [ABSTRACT FROM AUTHOR]

The zeta-functions are very interesting objects in analytic number theory. In the paper, the main attention is focused to the Riemann zeta-function, from the first results on the denseness until limit theorems with application in the investigation of universality in Voronin's sense. A special attention is devoted to the Matsumoto zeta-function. The paper is based on the content of a lecture to the master and doctoral students at the Graduate School of Mathematics of Nagoya University (Japan). [ABSTRACT FROM AUTHOR]

In this paper, limit theorems in the sense of the weak convergence of probability measures on the complex plane as well as in the space of analytic functions for the Laplace transform of the square of the Riemann zeta-function are obtained. [ABSTRACT FROM AUTHOR]

We consider the periodic zeta-function introduced by B. C. Berndt and L. Schoenfeld. We obtain the asymptotics of the mean square and prove limit theorems in the space of analytic functions for this function. [ABSTRACT FROM AUTHOR]

With the use of the boundedness condition, the asymptotics of negative moments of the normalized Riemann zeta-function is obtained. [ABSTRACT FROM AUTHOR]

In the paper, a Dirichlet series ζ u N (s) whose shifts ζ u N (s + i k h) , k = 0 , 1 , ⋯ , h > 0 , approximate analytic non-vanishing functions defined on the right-hand side of the critical strip is considered. This series is closely connected to the Riemann zeta-function. The sequence u N → ∞ and u N ≪ N 2 as N → ∞ . [ABSTRACT FROM AUTHOR]