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Let τ (n) be the Dirichlet divisor function and k ⩾ 2 be a fixed integer. We give an asymptotic formula of the mean square of Δ k (x) = ∑ n 1 , ⋯ , n k ⩽ x τ (n 1 ⋯ n k) − x k P k (log ⁡ x). [ABSTRACT FROM AUTHOR]

For q > 1, a new Green's function provides solutions of inhomogeneous multiplicatively advanced ordinary differential equations (iMADEs) of form y(N)(t) − Ay(qt) = f(t) for t ∈ [0, ∞). Such solutions are extended to global solutions on ℝ. Applications to inhomogeneous separable multiplicatively advanced partial differential equations are presented. Solutions to a linear free forced q‐advanced Schrödinger equation are obtained, opening an avenue to applications in quantum mechanics. New q‐Mittag‐Leffler functions qEα,β and ΥN,p govern the allowable decay rate of the inhomogeneities f(t) in the above iMADE. This provides a refinement to standard distribution theory, as we show is necessary for this study of iMADEs. A q‐Fredholm theory is developed and related to the above approach. For f(t) whose antiderivatives provide eigenfuntions of the noncompact integral operator K below, we exhibit solutions of the iMADE. Examples are provided, including a certain class of Dirichlet series. [ABSTRACT FROM AUTHOR]

The contemporary world is marked by generation and consumption of vast volume, high velocity, and considerable diverse data, leading us to the concept of big data. In this study, a system design thinking approach was employed to identify the requirements of Iran's public sector big data system. National big data systems would help governments to support their decisions by data and answer to national problems faster. Given the complexity and time-intensive nature of traditional system requirement analysis methods, their practical application in the industry has been declined. Therefore, in this research, system design thinking as an agile alternative for identifying system requirements has been discussed. To accomplish this, the LDA machine learning method has been utilized to analyze approximately 88,000 articles, a thematic analysis on around 600 Instagram and Twitter posts has been conducted, and six experts representing targeted problem persona were interviewed. The objective of this research is to extract insights to serve as a foundation for formulating big data policies in Iran. Findings reveal that Iran big data system requirements can be classified into four categories which indicate on increasing managed access to data while considering security and privacy, encouraging private and public sectors cooperation, transformation to smart governance, and establishing national data organization which would be responsible of data ID documents. [ABSTRACT FROM AUTHOR]

Let $ \tau(n) $ be the Dirichlet divisor function and $ k\geqslant2 $ be a fixed integer. We give an asymptotic formula of the mean square of$ \begin{equation*} \Delta_k(x) = \sum\limits_{n_1, \cdots, n_k\leqslant x}\tau(n_1 \cdots n_k)-x^kP_k(\log x). \end{equation*} $

For concrete, the strain tends to grow when the stress is kept at a constant level. This phenomenon is usually referred to as creep. Creep is an important physical property of concrete. For a prestressed concrete containment, creep could lead to prestress losses, stress redistribution, additional displacements, and even cracking. In general, the stress-strain relation of creep is nonlinear, but the principal stresses of concrete remain within the service stress range which is below 40% to 50% of the uniaxial strength. Therefore, the superposition principle can be utilized in linear elasticity, which works with the current values of stress and strain. Based on the theory of linear viscoelasticity, the principle of superposition can be used to characterize creep at a constant stress and the compliance function is used to describe the concrete creep mathematically, which facilitates numerical calculations. However, when the exponential algorithm is used to solve the creep effect of concrete, it is necessary to express the concrete creep compliance function by Dirichlet series and the calculation of the Dirichlet series corresponding to the compliance function is the key to implementing the exponential algorithm. The Weeks method for the inverse Laplace transform was used to approximate the Dirichlet series based on the continuous retardation spectrum method. The problem of approximating the concrete creep compliance function by the Weeks method was examined. First, the process of using the Weeks method to solve the continuous retardation spectrum was introduced. By taking the CEB MC90 creep model commonly used in engineering as an example, the equations for solving the concrete creep compliance function were derived by the Weeks method. The idea for improving the performance of the Weeks method was proposed. Based on this idea, the ranges of the various parameters that play a role in this solution were proposed. The numerical integration formula for the time-dependent term in the compliance function was derived. The results show that the calculation relative error with this method is no larger than ±1% when the duration is larger than 10 days. The validity of the algorithm was checked by comparing the numerical algorithm with the exact solution. This method is well suited for calculating the concrete creep compliance function for a long-term duration. The solution based on the Weeks method only requires the first-order derivative of the concrete creep compliance function to obtain the explicit function in the time domain, avoiding the complex computations of high-order derivatives and the low computational efficiency. Finally, the efficient Weeks method developed for the concrete creep model of CEB MC90 can also be extended and applied to other concrete creep models such as ACI 209R-92, JSCE, and GL2000.

The article "Extremal Positivity Problem for Integrals of Sine Series with Monotone Coefficients" discusses a theorem and problem related to sine series with monotone coefficients. The theorem establishes an inequality for sine series with monotone coefficients, leading to a posed problem to find the least upper bound for the inequality to hold. The article presents Theorem 1, which provides equalities and estimates for the problem, along with a detailed proof. The work was financially supported by the Russian Science Foundation and the authors declare no conflicts of interest. [Extracted from the article]

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We give necessary and sufficient conditions for a composition operator with Dirichlet series symbol to belong to the Schatten classes Sp of the Hardy space H² of Dirichlet series. For p≥2, these conditions lead to a characterization for the subclass of symbols with bounded imaginary parts. Finally, we establish a comparison-type principle for composition operators. Applying our techniques in conjunction with classical geometric function theory methods, we prove the analogue of the polygonal compactness theorem for H² and we give examples of bounded composition operators with Dirichlet series symbols on Hp, p > 0. [ABSTRACT FROM AUTHOR]

For the Dirichlet series F (s) = ∑ n = 1 ∞ f n exp { s λ n } , which is the Hadamard composition of the genus m of similar Dirichlet series F j (s) with the same exponents, the growth with respect to the function G (s) given as the Dirichlet series is studied in terms of the Φ -type (the upper limit of M G − 1 (M F (σ)) / Φ (σ) as σ ↑ A ) and convergence Φ -class defined by the condition ∫ σ 0 A Φ ′ (σ) M G − 1 (M F (σ)) Φ 2 (σ) d σ < + ∞ , where M F (σ) is the maximum modulus of the function F at an imaginary line and A is the abscissa of the absolute convergence. [ABSTRACT FROM AUTHOR]

We develop the L-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic twist of modular L-functions using multiple Dirichlet series under the generalized Riemann hypothesis. [ABSTRACT FROM AUTHOR]

In the paper, we consider simultaneous approximation of a pair of analytic functions by discrete shifts ζuN (s + ikh1; a) and ζuN (s + ikh2, α; b) of the absolutely convergent Dirichlet series connected to the periodic zeta-function with multiplicative sequence a, and the periodic Hurwitz zeta-function, respectively. We suppose that uN → ∞ and uN ≪ N2 as N → ∞, and the set {(h1 log p: p ∈ P), (h2 log(m + α): m ∈ N0), 2π} is linearly independent over Q. [ABSTRACT FROM AUTHOR]

The Lerch zeta-function L(λ, α, s), s = σ + it, depends on two real parameters P λ and 0 < α ⩽ 1, and, for σ > 1, is defined by the Dirichlet series ∞ m=0 e2πiλm(m + α)-s, and by analytic continuation elsewhere. In the paper, we consider the joint approximation of collections of analytic functions by discrete shifts (L(λ1, α1, s+ikh1), . . ., L(λr, αr, s+ikhr)), k = 0, 1, . . ., with arbitrary λj, 0 < αj ⩽ 1 and hj > 0, j = 1, . . ., r. We prove that there exists a non-empty closed set of analytic functions on the critical strip 1/2 < σ < 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. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be an integer. We studied the number of integers which form perfect k-th powers in the way of x(y1²+y2²+y3²+y4²)=zk. For k ≥ 4, we established a unified asymptotic formula with a power-saving error term for the number of such integers of bounded size under Lindelöf hypothesis, and we also gave an unconditional result for k = 2. [ABSTRACT FROM AUTHOR]

For a Dirichlet series of form F(s) = exp{sλ1} + P+∞ k=2 fk exp{sλk} (1) abcolutely convergent in the half-plane Π0 = {s: Re s < 0} new sufficient conditions for the close-to-pseudoconvexity are found and the obtained result is applied to studying of solutions linear differential equations of second order with exponential coefficients. In particular, are proved the following statements: 1) Let λk = λk-1 + λ1 and fk > 0 for all k ≥ 2. If 1 ≤ λ2f2/λ1 ≤ 2 and λkfk-λk+1fk+1 ↘ q ≥ 0 as k → +∞ then function of form (1) is close-to-pseudoconvex in Π0 (Theorem 3). This theorem complements Alexander's criterion obtained for power series. 2) If either -h2 ≤ γ ≤ 0 or γ = h2 then differential equation (1-ehs)2w′′-h(1-e2hs)w′+γe2hs = 0 (h > 0, γ ∈ R) has a solution w = F of form (1) with the exponents λk = kh and the the abscissa of absolute convergence σa = 0 that is close-to-pseudoconvex in Π0 (Theorem 4). [ABSTRACT FROM AUTHOR]

Let λ = (λn)n∈N0 be a non-negative sequence increasing to +∞, τ (λ) = limn→∞(ln n/λn), and D0(λ) be the class of all Dirichlet series of the form F(s) = P∞ n=0 an(F)esλn absolutely convergent in the half-plane Re s < 0 with an(F) ̸= 0 for at least one integer n ≥ 0. Also, let α be a continuous function on [x0,+∞) increasing to +∞, β be a continuous function on [a, 0) such that β(σ) → +∞ as σ ↑ 0, and γ be a continuous positive function on [b, 0). In the article, we investigate the growth of a Dirichlet series F ∈ D0(λ) depending on the behavior of the sequence (|an(F)|) in terms of its α, β, γ-orders determined by the equalities... where μ(σ) = max{|an(F)|eσλn: n ≥ 0} and M(σ) = sup{|F(s)|: Re s = σ} are the maximal term and the supremum modulus of the series F, respectively. In particular, if for every fixed t > 0 we have α(tx) ∼ α(x) as x → +∞, β(tσ) ∼ t-ρβ(σ) as σ ↑ 0 for some fixed ρ > 0, 0 < limσ↑0 γ(tσ)/γ(σ) ≤ limσ↑0 γ(tσ)/γ(σ) < +∞, Φ(σ) = α-1(β(σ))/γ(σ) for all σ ∈ [σ0, 0), eΦ(x) = max{xσ -Φ(σ): σ ∈ [σ0, 0)} for all x ∈ R, and ΔΦ(λ) = limn→∞(-ln n/eΦ(λn)), then: (a) for each Dirichlet series F ∈ D0(λ) we have... (b) if τ (λ) > 0, then for each p0 ∈ [0,+∞] and any positive function Ψ on [c, 0) there exists a Dirichlet series F ∈ D0(λ) such that R∗ α,β,γ(F) = p0 and M(σ, F) ≥ Ψ(σ) for all σ ∈ [σ0, 0); (c) if τ (λ) = 0, then (Rα,β,γ(F))1/ρ ≤ (R∗ α,β,γ(F))1/ρ + ΔΦ(λ) for every Dirichlet series F ∈ D0(λ); (d) if τ (λ) = 0, then for each p0 ∈ [0,+∞] there exists a Dirichlet series F ∈ D0(λ) such that R∗ α,β,γ(F) = p0 and (Rα,β,γ(F))1/ρ = (R∗ α,β,γ(F))1/ρ + ΔΦ(λ). [ABSTRACT FROM AUTHOR]

Let P be the set of generalized prime numbers, and ζ P (s) , s = σ + i t , denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζ P (s + i τ) , 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 { log p : p ∈ P } , and the existence of a bounded mean square for ζ P (s) . Under the above hypotheses, we obtain the universality of the function ζ P (s) . This means that the set of shifts ζ P (s + i τ) 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. [ABSTRACT FROM AUTHOR]

In this article, we have studied the uniqueness problem of Dirichlet series, which is convergent in a right half-plane and having analytic continuation in the complex plane as a meromorphic function sharing some sets and values. Our first result partially improve a result of [Ann. Univ. Sci. Budapest., Sect. Comput., 48(2018), 117-128] by relaxing the sharing conditions. Most importantly, we have pointed out a number of big gaps in a recent paper [J. Contemp. Math. Anal., 56(2021), 80-86], which makes the existence of the paper under question. Finally, under a different approach, we have provided the corrected form of the result of [J. Contemp. Math. Anal., 56(2021), 80-86] as much as practicable.

Creep is an important physical property of concrete and can lead to additional displacement, stress redistribution, and even cracking in concrete structures, inducing prestress loss of large-scale prestressed concrete structures. When an exponential algorithm is used to calculate the long-term creep of concrete, it is usually necessary to apply the continuous retardation spectra of the material. In the improved approach proposed here, the continuous retardation spectra can be obtained by the Weeks inverse Laplace transform. The CEB MC90 creep model is taken as an example to analyze the computational process, efficiency, and error of the approach. The improved approach is further applied to the ACI 209R-92, JSCE, and GL2000 concrete creep models. Through comparison with other methods, the advantages of the improved approach are illustrated, and some useful conclusions are drawn. [ABSTRACT FROM AUTHOR]

Featured Application: This paper describes the development of the constitutive model for early-aging viscoelastic concrete. And this paper provides the work related to material testing and modeling for identifying the required constitutive properties of early-age concrete. Short-term uniaxial tensile creep tests were conducted for early-age concrete at different ages in an effort to characterize early-age concrete as an aging viscoelastic material. Based on the test results, the viscoelastic material properties of the test concrete were characterized in terms of the Dirichlet series creep compliance and relaxation modulus functions. Applications of the model to the numerical analysis of the test samples showed a good agreement between the tests and computational results for the samples tested at different ages. This paper presents the test procedures and data analysis with a brief introduction of theoretical background. [ABSTRACT FROM AUTHOR]

The paper deals with the asymptotic behavior of the heat transfer in a bounded domain formed by two ε-periodically interwoven components, of highly different conductivities. Both components might be connected. At the interface, the heat flux is continuous and the temperature subjects to a first-order jump condition. The homogeneous Dirichlet condition is imposed on the exterior boundary. We determine the macroscopic law when the order of magnitude of the jump transmission coefficient is εr,-1 < r ≤ 1, using the two-scale convergence technique of the homogenization theory. [ABSTRACT FROM AUTHOR]

A Dirichlet series F(s) = ehs + ∑k=2∞ fkesλk with the exponents 0 < h < λk ↑ +∞ and the abscissa of absolute convergence σa[F] ≥ 0 is said to be pseudostarlike of order α ∈ [0, h) and type β ∈ (0, 1] in Π0 = {s : Re s < 0} if |F′(s)/F(s) − h| < β |F′(s)/F(s) − (2α − h)| for all s ∈ Π0. Similarly, the function F is said to be pseudoconvex of order α ∈ [0, h) and type β ∈ (0, 1] if |F′′(s)/F′(s) − h| < β |F′′(s)/F′(s) − (2α − h)| for all s ∈ Π0, and F is said to be close-to-pseudoconvex if there exists a pseudoconvex (with α = 0 and β = 1) function Ψ such that Re{F′(s)/Ψ′(s)} > 0 in Π0. Conditions on parameters a1, a2, b1, b2, c1, c2, under which the differential equation dnw/dsn + (a1ehs + a2) dw/ds + (b1 ehs + b2)w = c1 ehs + c2, n ≥ 2, has an entire solution pseudostarlike or pseudoconvex of order α ∈ [0, h) and type β ∈ (0, 1], or close-to-pseudoconvex in Π0 are found. It is proved that for such solution ln M(σ, F) = (1 + o(1)) n n√|b1 | /h ehσ/n as σ → +∞, where M(σ, F) = sup{|F(σ + it)| : t ∈ R}. [ABSTRACT FROM AUTHOR]

In the paper, using several approaches, the authors expand the composite function ln(1 + ex) into power series around x = 0, whose coefficients are expressed in terms of the Dirichlet eta function η(1 − n) and the Stirling numbers of the second kind S(n, k). [ABSTRACT FROM AUTHOR]

We consider the class S(λ, β, τ) of convergent for all x ≥ 0 Taylor-Dirichlet type series of the form F(x) = X+∞ n=0 bnexλn+τ(x)βn, bn ≥ 0 (n ≥ 0), where τ: [0,+∞) → (0,+∞) is a continuously differentiable non-decreasing function, λ = (λn) and β = (βn) are such that λn ≥ 0, βn ≥ 0 (n ≥ 0). In the paper we give a partial answer to a question formulated by Salo T.M., Skaskiv O.B., Trusevych O.M. on International conference "Complex Analysis and Related Topics" (Lviv, September 23-28, 2013) ([2]). We prove the following statement: For each increasing function h(x): [0,+∞) → (0,+∞), h′(x) ↗ +∞ (x → +∞), every sequence λ = (λn) such that X+∞ n=0 1 λn+1 - λn < +∞ and for any non-decreasing sequence β = (βn) such that βn+1 - βn ≤ λn+1 - λn (n ≥ 0) there exist a function τ (x) such that τ ′(x) ≥ 1 (x ≥ x0), a function F ∈ S(α, β, τ), a set E and a constant d > 0 such that h-meas E := R E dh(x) = +∞ and (∀x ∈ E): F(x) > (1 + d)μ(x, F), where μ(x, F) = max{|an|exλn+τ(x)βn: n ≥ 0} is the maximal term of the series. At the same time, we also pose some open questions and formulate one conjecture. [ABSTRACT FROM AUTHOR]

For h > 0, α ∈ [0, h) and μ ∈ R denote by SDh(μ, α) a class of absolutely convergent in the half-plane Π0 = {s: Re s < 0} Dirichlet series F(s) = esh + P∞ k=1 fk exp{sλk} such that Re n (μ-1)F′(s)-μF′′(s)/h (μ-1)F(s)-μF′(s)/h o > α for all s ∈ Π0, and let ΣDh(μ, α) be a class of absolutely convergent in half-plane Π0 Dirichlet series F(s) = e-sh + P∞ k=1 fk exp{sλk} such that Re n (μ-1)F′(s)+μF′′(s)/h (μ-1)F(s)+μF′(s)/h o < -α for all s ∈ Π0. Then SDh(0, α) consists of pseudostarlike functions of order α and SDh(1, α) consists of pseudoconvex functions of order α. For functions from the classes SDh(μ, α) and ΣDh(μ, α), estimates for the coefficients and growth estimates are obtained. In particular, it is proved the following statements: 1) In order that function F(s) = esh + P∞ k=1 fk exp{sλk} belongs to SDh(μ, α), it is sufficient, and in the case when fk(μλk/h - μ + 1) ≤ 0 for all k ≥ 1, it is necessary that ∞P k=1 fk 􀀀 μλk h - μ + 1 (λk - α) ≤ h - α, where h > 0, α ∈ [0, h) (Theorem 1). 2) In order that function F(s) = e-sh+ P∞ k=1 fk exp{sλk} belongs to ΣDh(μ, α), it is sufficient, and in the case when fk(μλk/h + μ - 1) ≤ 0 for all k ≥ 1, it is necessary that ∞P k=1 fk 􀀀 μλk h + μ - 1 (λk + α) ≤ h - α, where h > 0, α ∈ [0, h) (Theorem 2). Neighborhoods of such functions are investigated. Ordinary Hadamard compositions and Hadamard compositions of the genus m were also studied. [ABSTRACT FROM AUTHOR]

When the cells are ordinal in the multinomial distribution, i.e., when cells have a natural ordering, guaranteeing that the borrowing information among neighboring cells makes sense conceptually. In this paper, we introduce a novel probability distribution for borrowing information among neighboring cells in order to provide reliable estimates for cell probabilities. The proposed smoothed Dirichlet distribution forces the probabilities of neighboring cells to be closer to each other than under the standard Dirichlet distribution. Basic properties of the proposed distribution, including normalizing constant, moments, and marginal distributions, are developed. Sample generation of smoothed Dirichlet distribution is discussed using the acceptance-rejection algorithm. We demonstrate the performance of the proposed smoothed Dirichlet distribution using 2018 Major League Baseball (MLB) batters data. [ABSTRACT FROM AUTHOR]

Let p ∈ N, s=(s1,...,sp) ∈ Cp, h=(h1,...,hp) ∈ R+p, (n)=(n1,...,np) ∈ Np and the sequences λ(n)=(λn1(1),...,λnp(p)) are such that 0<λ(j)1<λk(j)<λk+1(j)↑+∞as k→∞ for every j=1,...,p. For a=(a1,...,ap) and c=(c1,...,cp) let (a,c)=a1c1+...+apcp, and we say that a>c if aj>cj for all 1≤j≤p. For a multiple Dirichlet series ... absolutely converges in Π0p = {s: Res < 0}, concepts of pseudostarlikeness and pseudoconvexity are introduced and criteria for pseudostarlikeness and the pseudoconvexity are proved. Using the obtained results, we investigated neighborhoods of multiple Dirichlet series, Hadamard compositions, and properties of solutions of some differential equations. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the sum of en- tire Dirichlet series with positive exponents on curves of a bounded slope going in a certain way to infinity. For entire transcendental functions of finite order, Polia showed that if the density of the sequence of exponents is equal to zero, then for any curve going to infinity there is an unbounded sequence of points on which the logarithm of the modulus of the sum of the series is equivalent to the logarithm of the maximum of the modulus. Later, these results were completely transferred by I. D. Latypov to entire Dirichlet series of finite order and finite lower order by Ritt. Further gener- alization was obtained in the works of N. N. Yusupova–Aitkuzhina to the more general dual classes of Dirichlet series defined by the convex majorant. In this paper, we obtain necessary and sufficient conditions for the exponents under which the logarithm of the mod- ulus of the sum of any Dirichlet series from one such class on a curve of bounded slope is equivalent to the logarithm of the maximum term on an asymptotic set whose upper density is not less than a positive number depending only on the curve.

For the Dirichlet series F(s)=∑n=1∞fnexp{sλn}, which is the Hadamard composition of the genus m of similar Dirichlet series Fj(s) with the same exponents, the growth with respect to the function G(s) given as the Dirichlet series is studied in terms of the Φ-type (the upper limit of MG−1(MF(σ))/Φ(σ) as σ↑A) and convergence Φ-class defined by the condition ∫σ0AΦ′(σ)MG−1(MF(σ))Φ2(σ)dσ<+∞, where MF(σ) is the maximum modulus of the function F at an imaginary line and A is the abscissa of the absolute convergence.

Let $F(s)=\sum\limits_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ and $F_j(s)=\sum\limits_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\},$ $j=\overline{1,p},$ be Dirichlet series with exponents $0\le\lambda_n\uparrow+\infty,$ $n\to\infty,$ and the abscissas of absolutely convergence equal to $0$. The function $F$ is called Hadamard composition of the genus $m\ge 1$ of the functions $F_j$ if $a_n=P(a_{n,1},\dots ,a_{n,p})$, where $$P(x_1,\dots ,x_p)=\sum\limits_{k_1+\dots+k_p=m}c_{k_1\dots\, k_p}x_1^{k_1}\cdots x_p^{k_p}$$ is a homogeneous polynomial of degree $m$. In terms of generalized orders and convergence classes the connection between the growth of the functions $F_j$ and the growth of the Hadamard composition $F$ of the genus $m\ge 1$ of $F_j$ is investigated. The pseudostarlikeness and pseudoconvexity of the Hadamard composition of the genus $m\ge 1$ are studied.

Let a = { a m : m ∈ N } be a periodic multiplicative sequence of complex numbers and L (s ; a) , s = σ + i t a Dirichlet series with coefficients a m . In the paper, we obtain a theorem on the approximation of non-vanishing analytic functions defined in the strip 1 / 2 < σ < 1 via discrete shifts L (s + i h t k ; a) , h > 0 , k ∈ N , where { t k : k ∈ N } is the sequence of Gram points. We prove that the set of such shifts approximating a given analytic function is infinite. This result extends and covers that of [Korolev, M.; Laurinčikas, A. A new application of the Gram points. Aequat. Math. 2019, 93, 859–873]. For the proof, a limit theorem on weakly convergent probability measures in the space of analytic functions is applied. [ABSTRACT FROM AUTHOR]

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]

Let ( λ n ) be a sequence of the pairwise distinct complex numbers. For a formal Dirichlet series F ( z ) = + ∞ ∑ n = 0 a n e z λ n, z ∈ C, we denote G μ ( F ), G c ( F ), G a ( F ) the domains of the existence, of the convergence and of the absolute convergence of maximal term μ ( z, F ) = max { | a n | e R ( z λ n ) : n ≥ 0 }, respectively. It is well known that G μ ( F ), G a ( F ) are convex domains. Let us denote N 1 ( z ) := { n : R ( z λ n ) > 0 }, N 2 ( z ) := { n : R ( z λ n ) < 0 } and α ( 1 ) ( θ ) := lim –––– n → + ∞ n ∈ N 1 ( e i θ ) − ln | a n | R ( e i θ λ n ), α ( 2 ) ( θ ) := ¯¯¯¯¯¯¯¯ lim n → + ∞ n ∈ N 2 ( e i θ ) − ln | a n | R ( e i θ λ n ) . Assume that a n → 0 as n → + ∞ . In the article, we prove the following statements. 1 ) If α ( 2 ) ( θ ) < α ( 1 ) ( θ ) for some θ ∈ [ 0, π ) then { t e i θ : t ∈ ( α ( 2 ) ( θ ), α ( 1 ) ( θ ) ) } ⊂ G μ ( F ) as well as { t e i θ : t ∈ ( − ∞, α ( 2 ) ( θ ) ) ∪ ( α ( 1 ) ( θ ), + ∞ ) } ∩ G μ ( F ) = ∅ . 2 ) G μ ( F ) = ⋃ θ ∈ [ 0, π ) { z = t e i θ : t ∈ ( α ( 2 ) ( θ ), α ( 1 ) ( θ ) ) } . 3 ) If h := lim –––– n → + ∞ − ln | a n | ln n ∈ ( 1, + ∞ ), then ( h h − 1 ⋅ G a ( F ) ) ⊃ G μ ( F ) ⊃ G c ( F ) . If h = + ∞ then G a ( F ) = G c ( F ) = G μ ( F ), therefore G c ( F ) is also a convex domain. [ABSTRACT FROM AUTHOR]

A ring R satisfies the right Beachy-Blair condition if for every faithful right ideal J of a ring R (that is, a right ideal J of a ring R is faithful if rR(J) = 0) is co-faithful (that is, a right ideal J of a ring R is called co-faithful if there exists a finite subset J1 ⊆ J such that rR(J1) = 0). In this note, we prove two main results. (a) Let R be a ring which is skew Hurwitz series-wise Armendariz, ω-compatible and torsion-free as a Z-module, and ω be an automorphism of R. If R satisfies the right Beachy-Blair condition then the skew Hurwitz series ring (HR, ω) satisfies the right Beachy-Blair condition. (b) Let MR be a right R-module which is ω-Armendariz of skew Hurwitz series type and torsion-free as a Z-module, and ω be an automorphism of R. If MR satisfies the right Beachy-Blair condition then the skew Hurwitz series module HM(HR, ω) satisfies the right Beachy-Blair condition. [ABSTRACT FROM AUTHOR]

We study stability of the maximal term of the Dirichlet series with positive exponents, the sum of which is an entire function. This problem is of interest, because the Leont’ev formulas for coefficients calculated using a biorthogonal system of functions play the key role in obtaining asymptotic estimates for entire Dirichlet series on various continua going to infinity (for example, curves). This fact naturally leads to the need to study the behavior of the logarithm of the maximum term also for the Hadamard composition of the corresponding Dirichlet series. For the wide class of entire Dirichlet series determined by a convex growth majorant, we establish a criterion for the equivalence of the logarithms of the moduli of the original series and a modified Dirichlet series outside some exceptional set.

In this paper we study the absolute convergence of general multiple Dirichlet series defined by ∑ m 1 = 1 ∞ ∑ m 2 = 1 ∞ ⋯ ∑ m r = 1 ∞ a 1 (m 1) a 2 (m 2) ⋯ a r (m r) m 1 s 1 (m 1 + m 2) s 2 ⋯ (m 1 + m 2 + ⋯ + m r) s r , where a j (1 ≤ j ≤ r) are arithmetic functions. In particular we completely determine the region of absolute convergence under certain conditions on the arithmetic functions. [ABSTRACT FROM AUTHOR]

Let a = { a m } and b = { b m } be two periodic sequences of complex numbers, and, additionally, a is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts (ζ n T (s + i τ ; a) , ζ n T (s + i τ , α ; b)) of absolutely convergent Dirichlet series ζ n T (s ; a) and ζ n T (s , α ; b) involving the sequences a and b is considered. Here, n T → ∞ and n T ≪ T 2 as T → ∞ . The coefficients of these series tend to a m and b m , respectively. It is proved that the set of the above shifts in the interval [ 0 , T ] has a positive density. This generalizes and extends the Mishou joint universality theorem for the Riemann and Hurwitz zeta-functions. [ABSTRACT FROM AUTHOR]

It is proved analogues of the classical Wiman's inequality for the class D of absolutely convergents in the whole complex plane Cp (entire) Dirichlet series of the form... with such a sequence of exponents... and... we denote... the sequence (-ln |an|)n∈Z p + arranged by non-decreasing. The main result of the paper: Let F ∈ D. If ... then there exists a set... such that... and relation... In general, under the specified conditions, the obtained inequality is exact. [ABSTRACT FROM AUTHOR]

A function ... with ... is called the Hadamard composition of the genus m ≥ 1 of functions ... where ... is a homogeneous polynomial of degree m ≥ 1. Let ... and functions α, β be positive continuous and increasing to +∞ on [x0,+∞). To characterize the growth of the function M(σ, F), we use generalized ... generalized type ... and membership in the converZgence class defined by the condition... Assuming the functions α, β and α-1(cβ(ln x)) are slowly increasing for each c ∈ (0,+∞) and ln n = O(λn) as n → ∞, it is proved, for example, that if the functions Fj have the same generalized order ϱα,β[Fj ] = ϱ ∈ (0,+∞) and the types Tα,β[Fj ] = Tj ∈ [0,+∞), cm0...0 = c ̸= 0, |an,1| > 0 and |an,j | = o(|an,1|) as n → ∞ for 2 ≤ j ≤ p, and F is the Hadamard composition of genus m ≥ 1 of the functions Fj then ϱα,β[F] = ϱ and ... It is proved also that F belongs to the generalized convergence class if and only if all functions Fj belong to the same convergence class. [ABSTRACT FROM AUTHOR]

Bayesian models based on the Dirichlet process and other stick-breaking priors have been proposed as core ingredients for clustering, topic modeling, and other unsupervised learning tasks. However, due to the flexibility of these models, the consequences of prior choices can be opaque. And so prior specification can be relatively difficult. At the same time, prior choice can have a substantial effect on posterior inferences. Thus, considerations of robustness need to go hand in hand with nonparametric modeling. In the current paper, we tackle this challenge by exploiting the fact that variational Bayesian methods, in addition to having computational advantages in fitting complex nonparametric models, also yield sensitivities with respect to parametric and nonparametric aspects of Bayesian models. In particular, we demonstrate how to assess the sensitivity of conclusions to the choice of concentration parameter and stick-breaking distribution for inferences under Dirichlet process mixtures and related mixture models. We provide both theoretical and empirical support for our variational approach to Bayesian sensitivity analysis. [ABSTRACT FROM AUTHOR]

The evolution of communities in dynamic (time-varying) network data is a prominent topic of interest. A popular approach to understanding these dynamic networks is to embed the dyadic relations into a latent metric space. While methods for clustering with this approach exist for dynamic networks, they all assume a static community structure. This paper presents a Bayesian nonparametric model for dynamic networks that can model networks with evolving community structures. Our model extends existing latent space approaches by explicitly modeling the additions, deletions, splits, and mergers of groups with a hierarchical Dirichlet process hidden Markov model. Our proposed approach, the hierarchical Dirichlet process latent position cluster model (HDP-LPCM), incorporates transitivity, models both individual and group level aspects of the data, and avoids the computationally expensive selection of the number of groups required by most popular methods. We provide a Markov chain Monte Carlo estimation algorithm and demonstrate its ability to detect evolving community structure in a network of military alliances during the Cold War and a narrative network constructed from the Game of Thrones television series. [ABSTRACT FROM AUTHOR]

We study the relation between the composition operators Cϕ with closed range on the weighted Bloch spaces and Cϕ with closed range on the weighted Dirichlet spaces Dαp. In particular, we study the boundedness from below of composition operators between Lpa and Lqa, between Lpa and Hardy space, and between Lpa and Besov space. [ABSTRACT FROM AUTHOR]

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series. [ABSTRACT FROM AUTHOR]

For a sequence w = {wj}j=2∞ of positive real numbers, consider the positive semi-definite kernel kw(s,u) = ...defined on some right-half plane ... for a real number Ρ. Let Hw denote the reproducing kernel Hilbert space associated with KW. Let ... where [Pj}j≥1 is an increasing enumeration of prime numbers and gpf(n) denotes the greatest prime factor of an integer n ≥ 2. If w satisfies ... where µ is the Möbius function, then the multiplier algebra M(Hw of Hw is isometrically isomorphic to the space of all bounded and holomorphic functions on ... that are representable by a convergent Dirichlet series in some right half plane. As a consequence, we describe the multiplier algebra M(Hw) when w is an additive function satisfying δw ≤ 0 and ... for all integers j ≥ 2 and all prime numbers p. Moreover, we recover a result of Stetler that describes the multipliers of Hw when w is multiplicative. The proof of the main result is a refinement of the techniques of Stetler. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]