Search Results (10)

The aim of this work was to construct explicit expressions for the summation of Dirichlet Beta functions with odd arguments and Zeta functions with even arguments. In the established literature, this is typically done using Fourier series expansions or Bernoulli numbers and polynomials. Here, instead, we achieve our goal by employing tools from probability: specifically, we introduce a generalisation of a technique based on multiple integrals and the algebra of random variables. This also allows us to increase the number of nested integrals and Cauchy random variables involved. Another key contribution is that, by generalising the exponent of Cauchy random variables, we obtain an original proof of the reflection formulae for the Digamma and Trigamma functions. These probabilistic proofs crucially utilise the Mellin transform to compute the integrals needed to determine probability density functions. It is noteworthy that, while understanding the presented topic requires knowledge of the rules for calculating multiple integrals (Fubini’s Theorem) and the algebra of continuous random variables, these are concepts commonly acquired by second-year university students in STEM disciplines. Our study thus offers new perspectives on how the mathematical functions considered relate and shows the significant role of probabilistic methods in promoting comprehension of this research area, in a way accessible to a broad and non-specialist audience. Full article

This article considers a class of entire functions of several complex variables that are bounded in the Cartesian product of some half-planes. Each such hyperplane is defined on the condition that the real part of the corresponding variable is less than some r. For this class of functions, there are established analogs of the Wiman theorems. The first result describes the behavior of an entire function from the given class at the neighborhood of the point of the supremum of its modulus. The second result shows asymptotic equality for supremums of the modulus of the function and its real part outside some exceptional set. In addition, the analogs of Wiman’s theorem are obtained for entire multiple Dirichlet series with arbitrary non-negative exponents. These results are obtained as consequences of a new statement describing the behavior of an entire function $F\left(z\right)$ of several complex variables $z=(z_1,\dots ,z_p)$ at the neighborhood of a point w, where the value $F\left(w\right)$ is close to the supremum of its modulus on the boundary of polylinear domains. The paper has two moments of novelty: the results use a more general geometric exhaustion of p-dimensional complex space by polylinear domains than previously known; another aspect of novelty concerns the results obtained for entire multiple Dirichlet series. There is no restriction that every component of exponents is strictly increasing. These statements are valid for any non-negative exponents. Full article

New estimates for the convergence abscissas of the multiple Laplace–Stieltjes integral are obtained. There is described the relationship between the integrand function, the Lebesgue–Stieltjes measure, and the abscissa of convergence of the multiple Laplace–Stieltjes integral. Since the multiple Laplace–Stieltjes integral is a direct generalization of the Laplace integral and multiple Dirichlet series, known results about convergence domains for the multiple Dirichlet series are obtained as corollaries of the presented more general statements for the multiple Laplace–Stieltjes integral. Full article

Zeta-functions play a fundamental role in many fields where there is a norm or a means to measure distance. They are usually given in the forms of Dirichlet series (additive), and they sometimes possess the Euler product (multiplicative) when the domain in question has a unique factorization property. In applied disciplines, those zeta-functions which satisfy the functional equation but do not have Euler products often appear as a lattice zeta-function or an Epstein zeta-function. In this paper, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta-functions by considering some generalizations of the Eisenstein series of level $2\varkappa$ to the complex variable level s. Naturally, generalized Eisenstein series and Barnes multiple zeta-functions arise, which have affinity to dissections, as they are (semi-) lattice functions. The method of Lewittes (and Chapman) and Kurokawa leads to some limit formulas without absolute value due to Tsukada and others. On the other hand, Komori, Matsumoto and Tsumura make use of the Barnes multiple zeta-functions, proving their modular relation, and they give rise to generalizations of Ramanujan’s formula as the generating zeta-function of ${\sigma }_s\left(n\right)$, the sum-of-divisors function. Lewittes proves similar results for the 2-dimensional case, which holds for all values of s. This in turn implies the eta-transformation formula as the extreme case, and most of the results of Chapman. We shall unify most of these as a tapestry of ideas arising from the merging of additive entity (Dirichlet series) and multiplicative entity (Euler product), especially in the case of limit formulas. Full article

We consider a wide class of summatory functions $F\left\{f;N,p^m\right\}={\sum }_{k\le N}f\left(p^{m}k\right)$, $m\in {\mathbb{Z}}_{+}\cup \left\{0\right\}$ associated with the multiplicative arithmetic functions f of a scaled variable $k\in {\mathbb{Z}}_{+}$, where p is a prime number. Assuming an asymptotic behavior of the summatory function, $F\{f;N,1\}\stackrel{N\to \infty }{=}{G}_1\left(N\right)\left[1+\mathcal{O}\left(G_2\left(N\right)\right)\right]$, where $G_1\left(N\right)=N^{a_1}{\left(\log N\right)}^{b_1}$, $G_2\left(N\right)=N^{-a_2}{\left(\log N\right)}^{-b_2}$ and $a_1,a_2\ge 0$, $-\infty <b_1,b_2<\infty$, we calculate the renormalization function $R\left(f;N,p^m\right)$, defined as a ratio $F\left\{f;N,p^m\right\}/F\{f;N,1\}$, and find its asymptotics $R_{\infty }\left(f;p^m\right)$ when $N\to \infty$. We prove that a renormalization function is multiplicative, i.e., $R_{\infty }\left(f;{\prod }_{i=1}^n{p}_i^{m_i}\right)={\prod }_{i=1}^n{R}_{\infty }\left(f;p_i^{m_i}\right)$ with n distinct primes $p_i$. We extend these results to the other summatory functions ${\sum }_{k\le N}f\left(p^m{k}^l\right)$, $m,l,k\in {\mathbb{Z}}_{+}$ and ${\sum }_{k\le N}{\prod }_{i=1}^n{f}_i\left(k{p}^{m_i}\right)$, $f_i\ne f_j$, $m_i\ne m_j$. We apply the derived formulas to a large number of basic summatory functions including the Euler $\varphi \left(k\right)$ and Dedekind $\psi \left(k\right)$ totient functions, divisor ${\sigma }_n\left(k\right)$ and prime divisor $\beta \left(k\right)$ functions, the Ramanujan sum $C_q\left(n\right)$ and Ramanujan $\tau$ Dirichlet series, and others. Full article

Let $\mathfrak{a}=\{a_m:m\in \mathbb{N}\}$ be a periodic multiplicative sequence of complex numbers and $L(s;\mathfrak{a})$, $s=\sigma +it$ 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 $1/2<\sigma <1$ via discrete shifts $L(s+ih{t}_k;\mathfrak{a})$, $h>0$, $k\in \mathbb{N}$, where $\{t_k:k\in \mathbb{N}\}$ 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. Full article

Let $\mathfrak{a}=\left\{a_m\right\}$ and $\mathfrak{b}=\left\{b_m\right\}$ be two periodic sequences of complex numbers, and, additionally, $\mathfrak{a}$ is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts $({\zeta }_{n_T}(s+i\tau ;\mathfrak{a}),{\zeta }_{n_T}(s+i\tau ,\alpha ;\mathfrak{b}))$ of absolutely convergent Dirichlet series ${\zeta }_{n_T}(s;\mathfrak{a})$ and ${\zeta }_{n_T}(s,\alpha ;\mathfrak{b})$ involving the sequences $\mathfrak{a}$ and $\mathfrak{b}$ is considered. Here, $n_T\to \infty$ and $n_T\ll T^2$ as $T\to \infty$. 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. Full article

The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class $\mathcal{S}$ is a complemented Selberg class by an arithmetic hypothesis related to the distribution of prime numbers. In this paper, a joint universality theorem for the functions L from the class $\mathcal{S}$ on the approximation of a collection of analytic functions by shifts $\left(L(s+i{a}_1\tau ),\dots ,L(s+i{a}_r\tau )\right)$, where $a_1,\dots ,a_r$ are real algebraic numbers linearly independent over the field of rational numbers, is obtained. It is proved that the set of the above approximating shifts is infinite, its lower density and, with some exception, density are positive. For the proof, a probabilistic method based on weak convergence of probability measures in the space of analytic functions is applied together with the Backer theorem on linear forms of logarithms and the Mergelyan theorem on approximation of analytic functions by polynomials. Full article

In this paper, we consider the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems ${\Delta }_1^{2}u(i-1,j)+{\Delta }_2^{2}u(i,j-1)=-f((i,j),u(i,j)),(i,j)\in \Omega ,u(i,0)=u(i,T_2+1)=0i\in \mathbb{Z}(1,T_1),u(0,j)=u(T_1+1,j)=0j\in \mathbb{Z}(1,T_2),$ which have a symmetric structure. When the nonlinearity $f(·,u)$ is resonant at both zero and infinity, we construct a variational functional on a suitable function space and turn the problem of finding nontrivial solutions of discrete elliptic Dirichlet problems to seeking nontrivial critical points of the corresponding functional. We establish a series of results based on the existence of one, two or five nontrivial solutions under reasonable assumptions. Our results depend on the Morse theory and local linking. Full article

In this study, we present a state-based diagnostic and prognostic methodology for lubricating oil degradation based on a nonparametric Bayesian approach, i.e., sticky hierarchical Dirichlet process–hidden Markov model (HDP-HMM). An accurate health state-space assessment for diagnostics and prognostics has always been unobservable and hypothetical in the past. The lubrication condition monitoring (LCM) data is generally segregated as “healthy or unhealthy”, representing a binary state-based perspective to the problem. This two-state performance-based formulation poses limitations to the precision and accuracy of the diagnosis and prognosis for real data wherein there may be multiple states of discrete performance that are characteristic of the system functionality. In particular, the reversible and nonlinear time-series trends of degradation data increase the complexity of state-based modeling. We propose a multistate diagnostic and prognostic framework for LCM data in the wear-out phase (i.e., the unhealthy portion of degradation data), accounting for irregular oil replenishment and oil change effects (i.e., nonlinearity in the degradation signal). The LCM data is simulated for an elementary mechanical system with four components. The sticky HDP sets the prior for the HMM parameters. The unsupervised learning over infinite observations and emission reveals four discrete health states and helps estimate the associated state transition probabilities. The inferred state sequence provides information relating to the state dynamics, which provides further guidance to maintenance decision making. The decision making is further backed by prognostics based on the conditional reliability function and mean residual life estimation. Full article