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We give an axiomatic characterization of multiple Dirichlet series over the function field F q (T) , generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature. [ABSTRACT FROM AUTHOR]

Let d (1) (n) be the n -th coefficient of the Dirichlet series (ζ ′ (s)) 2 = ∑ n = 1 ∞ d (1) (n) n − s in ℜ s > 1 , and Δ (1) (x) be the error term of ∑ n ≤ x d (1) (n). In this paper, we will study the fifth-power moment of Δ (1) (x). [ABSTRACT FROM AUTHOR]

We evaluate asymptotically the smoothed first moment of central values of families of primitive cubic, quartic and sextic Dirichlet L -functions, using the method of double Dirichlet series. Quantitative non-vanishing results for these L -values are also proved. [ABSTRACT FROM AUTHOR]

Let π be a cuspidal unitary representation od G L (m , A) where A denotes the ring of adèles of Q. Let L (s , π) be its L -function. We introduce a universal lower bound for the integral ∫ − ∞ + ∞ | L (1 2 + i t , π) 1 2 + i t − s | 2 d t where s is equal to 0 or is a zero of L (s) on the critical line. In the main text, the proof is given for m ≤ 2 and under a few assumptions on π. It relies on the Mellin transform; the proof involves an extension of a deep result of Friedlander-Iwaniec. An application is given to the abscissa of convergence of the Dirichlet series L (s , π). In the Appendix, written with Peter Sarnak, the proof is made unconditional for general m. [ABSTRACT FROM AUTHOR]

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]

In this paper, we show that any Dirichlet series ∑ n = 1 ∞ f (n) n s , where s = σ + i t ∈ C , | f (n) | = 1 for all n ≥ 1 , and f (1) = 1 , has the property that a n n > - log 2 for n > 1 , where a n = inf ℜ s : S n (s) = 0 and S n (s) : = ∑ k = 1 n f (k) k s . Furthermore, if f is completely multiplicative, then - log 2 = lim n → ∞ a n n . [ABSTRACT FROM AUTHOR]

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Let r ⩾ 3 r\geqslant 3 be an integer and 푄 any positive definite quadratic form in 푟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q S_{Q} defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z x^{3}=Q(y_{1},\dots,y_{r})z . This confirms Manin’s conjecture for any S Q S_{Q} . Our proof is based on analytic methods, and uses some estimates for character sums and moments of 퐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3 . [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]

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function σ α (n) : = ∑ d | n d α. We obtain an exact identity relating the Dirichlet series ζ (s) ζ (s − α) and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip. [ABSTRACT FROM AUTHOR]

Let Λ = (λ n) be an increasing to +∞ sequence of non-negative numbers, λ0 = 0, and by S(Λ, A) we denote a class of Dirichlet series F s = ∑ n = 0 ∞ f n exp s λ n , s = σ + i t , with the abscissa of the absolute convergence σa = A ∈ (−∞,+∞]. For σ < A we put MF (σ) = sup{|F(σ + it)| : t ∈ ℝ}. The growth of the function F ∈ S(Λ,+∞) with respect to the function G s = ∑ n = 0 ∞ g n exp s λ n ∈ S Λ , 0 is identified with the growth of the function 1 / M G - 1 M F σ | as σ → +∞. In terms of generalized orders, the connection between the growth of this function and the behavior of the coefficients fn and gn has been studied. [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]

Let $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$ , we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $ [ABSTRACT FROM AUTHOR]

We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series. [ABSTRACT FROM AUTHOR]

In the paper 'Composition operators on weighted Bergman spaces of Dirichlet series. J Math Anal Appl. 2015;426:340–363', Bailleul completely characterized the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case of symbols with $ c_0\ge 1 $ c 0 ≥ 1. But the sufficient conditions for the other case $ c_0=0 $ c 0 = 0 were unsolved. In this paper, we follow this line and study the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case $ c_0=0 $ c 0 = 0. Moreover, we also obtain the compact characterizations of composition operators with $ c_0\geq 1 $ c 0 ≥ 1. [ABSTRACT FROM AUTHOR]

This article aims to achieve exact and analytical solutions for the classical Falkner–Skan equation with heat transfer (FSE‐HT). Specifically, when the pressure gradient parameter β=−1 $\beta =-1$, there already exists a closed‐form solution in the literature for the Falkner–Skan flow equation. The main purpose here is to extend this case to obtain a closed‐form solution to the heat transport equation with the solubility condition Pr=1 $Pr=1$. An algorithm is presented and is found to be new to the literature that enriches the physical properties of FSE‐HT. It is shown that for the moving wedge parameter λ>1 $\lambda \gt 1$, the momentum and temperature equations show multiple solutions analytically. The skin friction coefficient and the heat transfer rate are also obtained in analytical form. The thus‐obtained solution is then adapted to derive an analytical solution applicable to a wide range of pressure gradient parameters β $\beta $ and Prandtl numbers Pr $Pr$. Furthermore, an asymptotic analysis is conducted, focusing on scenarios where the moving wedge parameter becomes significantly large (λ→∞ $\lambda \to \infty $). Nevertheless, in all the above‐mentioned cases, the skin friction coefficient (f″(0) $f^{\prime\prime} (0)$) and the heat transfer rate (θ′(0) $\theta ^{\prime} (0)$) are compared with the direct numerical solutions of the boundary layer equations, and it is found that the results are in good agreement. These solutions provide a benchmark and shed light on further studies on the families of FSE‐HT. [ABSTRACT FROM AUTHOR]

We study the behavior of the shifted convolution sum involving even power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the k -full kernel function for any fixed integer k ≥ 2. [ABSTRACT FROM AUTHOR]

Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions, including moments or statistics about the distribution of zeros. We consider the family of real Dirichlet characters, and use multiple Dirichlet series to prove the ratios conjectures with one shift in the numerator and denominator in some range of the shifts. This range can be improved by extending the family to include non-primitive characters. All of the results are conditional under the Generalized Riemann hypothesis. This extended range is good enough to enable us to compute an asymptotic formula for the sum of shifted logarithmic derivatives near the critical line. As an application, we compute the one-level density for test functions whose Fourier transform is supported in \left (-2,2\right), including lower-order terms. [ABSTRACT FROM AUTHOR]

We obtain an asymptotic formula for the smoothly weighted first moment of quadratic Dirichlet L-functions at central values, with explicit main terms and an error term that is "square-root" of the main term. [ABSTRACT FROM AUTHOR]

For a natural number n , let Z 1 (n) : = ∑ ρ n ρ ρ where the sum runs over the nontrivial zeros of the Riemann zeta function. For a primitive Dirichlet character χ modulo q ≥ 3 , we define Z 1 (s , χ) : = ∑ n = 1 ∞ χ (n) Z 1 (n) n s for ℜ (s) > 2 and obtain the meromorphic continuation of the function Z 1 (s , χ) to the region ℜ (s) > 1 2 . Our main result indicates that the poles of Z 1 (s , χ) in the region 1 2 < ℜ (s) < 1 , if they exist, are related to the zeros of many Dirichlet L -functions in the same region. [ABSTRACT FROM AUTHOR]

A method to calculate partial sums of some multiple numerical series arising when searching for the resultant of a polynomial and an entire function is proposed. One can apply a symbolic algorithm that uses recurrent Newton formulas to find power sums of roots included in this formula without finding the very roots of the system. The algorithm that implements the proposed approach to calculate partial sums of multiple numerical series is implemented in Maple. Examples of using this algorithm to find partial sums of some classes of multiple numerical series are given. [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]

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 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]

The complex symmetric linear combinations of composition operators on the McCarthy–Bergman spaces of Dirichlet series are completely characterized. The normality and self-adjointness of complex symmetric linear combinations of composition operators on such spaces are also characterized. Some images are given in order to find some interesting phenomena of J -symmetric such combinations. [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]

Let B N be the Euclidean ball of C N . The space H ∞ (B N) of bounded holomorphic functions on B N is known to have a predual, denoted by G ∞ (B N) . We study the functions in H ∞ (B N) that attain their norm as elements of the dual of G ∞ (B N) . We also examine similar questions for the polydisc algebra H ∞ (D N) and for the space of Dirichlet series D ∞ (C +). [ABSTRACT FROM AUTHOR]

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 following extension of Bohr's theorem is established: If a somewhere convergent Dirichlet series f has an analytic continuation to the half-plane \mathbb {C}_\theta = \{s = \sigma +it\,:\, \sigma >\theta \} that maps \mathbb {C}_\theta to \mathbb {C} \setminus \{\alpha,\beta \} for complex numbers \alpha \neq \beta, then f converges uniformly in \mathbb {C}_{\theta +\varepsilon } for any \varepsilon >0. The extension is optimal in the sense that the assertion no longer holds should \mathbb {C}\setminus \{\alpha,\beta \} be replaced with \mathbb {C}\setminus \{\alpha \}. [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]

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]

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]

The article considers the behavior of the sum of the Dirichlet series which converges absolutely in the left half-plane , on a curve arbitrarily approaching the imaginary axis—the boundary of this half-plane. We have obtained a solution to the following problem: under what additional conditions on will the strengthened asymptotic relation the type of Pólya for the sum F(s) of the Dirichlet series be valid in the case when the argument tends to the imaginary axis along over a sufficiently massive set. [ABSTRACT FROM AUTHOR]

We give a sufficient condition for a composition operator with positive characteristic to be compact on the Hardy space of Dirichlet series. [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]

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.

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.

Ocean acoustic tomography (OAT) deploys most moored stations on the periphery of the tomographic region to sense the solenoidal current field. Moving vehicle tomography (MVT), an advancement of OAT, not only samples the region from various angles for improving the resolution of mapped currents but also acquires information about the irrotational flow due to the sampling points inside the region. To reconstruct a complete two-dimensional current field, the spatial modes derived from the open-boundary modal analysis (OMA) are preferable to the conventional truncated Fourier series since the OMA technique describes the solenoidal and irrotational flows efficiently in which all modes satisfy the coastline and open boundary conditions. Comparisons of the reconstructions are presented using three different representations of currents. The first two representations explain only the solenoidal flow: the truncated Fourier series and the OMA Dirichlet modes. The third representation, accounting for the solenoidal and irrotational flows, uses all the OMA modes. For reconstructing the solenoidal flow, the OMA representation with the Dirichlet modes performs better than the Fourier series. A large difference appears near the bay mouth, where the OMA-Dirichlet reconstruction shows a better fit to the uniform currents. However, considerable uncertainty exists outside the bay mouth where the irrotational currents dominate. This can be improved by the third representation with the inclusion of the Neumann and boundary modes. The reconstruction results using field data were validated against the acoustic Doppler current profiler (ADCP) measurements. Additionally, incorporating constraints from ADCP measurements enhances the accuracy of the reconstruction. Significance Statement: This study contributes toward improving our understanding of accurately measuring oceanic circulation patterns over large areas without relying solely upon stationary sensors or satellite imagery. The study combines multiple sources, such as shipboard ADCP and tomographic techniques, to obtain a complete picture of what is happening beneath surface waters across entire regions under investigation. It has important implications for fields such as climate science, marine biology, and fisheries management, where accurate knowledge of the movement and distribution of water masses is crucial for predicting future trends and making informed decisions. [ABSTRACT FROM AUTHOR]

Weyl group multiple Dirichlet series are Dirichlet series in r complex variables, with analytic continuation to C r and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank r. Such series may be defined for any global field K , but in the case when K is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when K = F q (T) , and describe the denominators and support of the numerators. [ABSTRACT FROM AUTHOR]

We study the number of two‐dimensional sublattices of Z4$\mathbb {Z}^4$ of a fixed covolume and construct the associated Dirichlet series. The latter is shown to be related to Eisenstein series on both GL4$\operatorname{GL}_4$ and its metaplectic double cover. [ABSTRACT FROM AUTHOR]

Haar and Kolmogorov found the necessary and sufficient conditions under which finite-dimensional subspaces in the space of continuous functions on an arbitrary compact set are Chebyshev. In this paper, it is proved that subspaces of Dirichlet series form Chebyshev subspaces in the space of of continuous and bounded functions in the interval that have a limit at infinity. [ABSTRACT FROM AUTHOR]

For a matrix a = (a m , n) m , n = 1 ∞ with scalar entries, the Dirichlet series kernel κ a is the double Dirichlet series κ a (s , u) = ∑ m , n = 1 ∞ a m , n m - s n - u ¯ in the variables s and u ¯ , which is regularly convergent on some right half-plane H ρ. The analytic symbols A n , a = ∑ m = 1 ∞ a m , n m - s , n ⩾ 1 play a central role in the study of the reproducing kernel Hilbert space H a associated with the positive semi-definite kernel κ a. In particular, they form a total subset of H a and provide the formula ∑ n = 1 ∞ ⟨ f , A n , a ⟩ n - s , s ∈ H ρ , for f ∈ H a. We combine the basic theory of Dirichlet series kernels with the Gelfond-Schneider theorem (Hilbert’s seventh problem) to show that any quasi-invariant Dirichlet series kernel κ a (s , u) factors as f (s) f (u) ¯ for some Dirichlet series f on H ρ. In particular, there is no quasi-invariant Dirichlet series kernel κ a if the dimension of H a is bigger than one. This is in strict contrast with the case of the unit disc, where non-factorable quasi-invariant kernels exist in abundance. We further discuss the Dirichlet series kernels κ a invariant under the group T of translation automorphisms of H ρ and construct a family of densely defined T -homogeneous operators in H a , whose adjoints are defined only at the zero vector. [ABSTRACT FROM AUTHOR]

We establish a subconvexity bound for a double Dirichlet series involving with the quadratic Hecke L -functions over the Gaussian field. [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]

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]