Abstract: Abstract We study the distribution of the discriminant D(P) of polynomials P from the class Pn(Q) of all integer polynomials of degree n and height at most Q. We evaluate the asymptotic number of polynomials P ∈ Pn(Q) having all real roots and satisfying the inequality D(P) ≤ X as Q→∞and X/Q2n−2→ 0. PubDate: 2019-03-06 DOI: 10.1007/s10986-019-09434-z
Abstract: Abstract We investigate the mixed joint discrete value distribution and the mixed joint discrete universality for the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with transcendental parameter. The common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are used. PubDate: 2019-03-05 DOI: 10.1007/s10986-019-09432-1
Abstract: Abstract For 0 < α ≤ 1 and 0 < λ ≤ 1, let L(α, λ, s) denote the associated Lerch zeta-function, which is defined as \( {\sum}_{n=0}^{\infty }{\mathrm{e}}^{2\pi \mathrm{i}n\uplambda}{\left(n+\alpha \right)}^{-s} \) for ℜs > 1. We investigate the joint value-distribution for all Lerch zeta-functions {L(αj, λ, s): 0 < λ ≤ 1, j = 1, … , J} and their derivatives when α1,…, αJ satisfy a certain condition. This condition is satisfied if α1,…, αJ are algebraically independent over ℚ. More precisely, we establish a joint denseness result for values of those functions on vertical lines in the strip 1/2 < ℜs ≤ 1. We also establish the functional independence of those functions in the sense of Voronin. PubDate: 2019-03-05 DOI: 10.1007/s10986-019-09433-0
Abstract: Abstract The triplet (a, b, c) of positive integers is said to be compositum-feasible if there exist number fields K and L of degrees a and b, respectively, such that the degree of their compositum KL equals c. We determine all compositum-feasible triplets (a, b, c) satisfying a ≤ b and b ∈ {8, 9}. This extends the classification of compositum-feasible triplets started by Drungilas, Dubickas, and Smyth [5]. Moreover, we obtain several results related to triplets of the form (a, a, c). In particular, we prove that the triplet (n, n, n(n − 2)) is not compositum-feasible, provided that n ≥ 5 is an odd integer. PubDate: 2019-03-01 DOI: 10.1007/s10986-019-09428-x
Abstract: Abstract We prove that any beta distribution can be simulated by means of a sequence of distributions defined via multiplicative functions in a short interval. PubDate: 2019-02-26 DOI: 10.1007/s10986-019-09427-y
Abstract: Abstract We consider simultaneous rational approximations to real and p-adic numbers. We prove that for any irrational number α0 and p-adic number α, there are infinitely many irreducible fractions satisfying ∣α0 − m/n ∣ < 1/n2/3 − ∈ and α0 − m/n p < 1/n2/3 − ∈, where ∈ > 0 is an arbitrary number, and ∣ ⋅ ∣ p is the p-adic norm. PubDate: 2019-02-23 DOI: 10.1007/s10986-019-09426-z
Abstract: Abstract Bohr and Jessen proved the existence of a certain limit value regarded as the probability that values of the Riemann zeta function belong to a given region in the complex plane. They also studied the density of the probability, which has been called the M-function since the studies of Ihara and Matsumoto. In this paper, we construct M-functions for the value-distributions of L-functions in a class containing many kinds of zeta and L-functions. Moreover, we improve the estimate on the rate of the convergence of the limit studied by Bohr and Jessen. PubDate: 2019-02-23 DOI: 10.1007/s10986-019-09425-0
Abstract: Abstract Suppose that an algebraic number β of degree d = n(n − 1) over ℚ is expressible by the difference of two conjugate algebraic integers α1 ≠ α2 of degree n, namely, β = α1− α2. We prove that then there exists a constant c > 1, which depends on \( \overline{\mid \alpha \mid } \) = max1≤i≤n ∣αi∣ only, such that M(β)1/d > c. PubDate: 2019-02-19 DOI: 10.1007/s10986-019-09424-1
Abstract: Abstract The paper extends the study of the modified Borwein method for the calculation of the Riemann zeta-function. We present an alternative perspective on the proof of the central limit theorem for coefficients of the method. The new approach is based on the connection with the limit theorem applied to asymptotic enumeration. PubDate: 2019-02-18 DOI: 10.1007/s10986-019-09421-4
Abstract: Abstract Let Q1, . . . , Qt ∈ℝ[x] be polynomials with no constant term for which each linear combination m1Q1(x) + · · ·+mtQt (x) with m1, . . . , mt ∈ℤand not all 0 always has an irrational coefficient. Let I1, . . . , It be sets included in the interval [0, 1), each being a union of finitely many subintervals of [0, 1). Furthermore, let \( \mathcal{T} \) be the set of positive integers n for which {Q1(n)} ∈ I1, . . . , {Qt(n)} ∈ It simultaneously, where {y} stands for the fractional part of y. Let t1, t2, . . . be a uniformly summable sequence of complex numbers and set T(x) = ∑ n ≤ xtn and \( T\left(x \mathcal{T}\right)=\sum {}_{n\le x,n\in \mathcal{T}}{t}_n \) . We prove that, as x→∞, \( T(x)/x\sim T\left(x \ \mathcal{T}\right)/\left(\uplambda \left({I}_1\right)\cdots \uplambda \left({I}_t\right)x\right) \) , where λ(I) is the Lebesgue measure of a set I. PubDate: 2019-02-18 DOI: 10.1007/s10986-019-09422-3
Abstract: Abstract 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}. PubDate: 2019-02-18 DOI: 10.1007/s10986-019-09423-2
Abstract: Abstract The classical Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts of the Riemann zeta-function ζ(s + iτ), τ ∈ ℝ. In the paper, we generalize this approximation for shifts ζ(s + iφ(τ)), where ðœ‘(τ) has the monotonic positive derivative such that 1/ðœ‘′(τ) = o(τ) and φ(2τ) × maxτ ≪ t ≪ 2τ(1/φ′(t)) ≪ τ as τ →∞. PubDate: 2019-01-18 DOI: 10.1007/s10986-019-09418-z
Abstract: Abstract In this paper, we extend tools developed in [26] to study harmonic numbers sums. We use these objects to compute Euler-related sums and to explore their connections with classical Euler sums. We establish some new relations between Euler sums and related sums. The relations obtained allow us to find some nice closed-form representations of nonlinear Euler sums through zeta values and linear sums. PubDate: 2019-01-18 DOI: 10.1007/s10986-019-09420-5
Abstract: Abstract In this work, we consider a second-order coupled system of differential equations in semiinfinite intervals. The arguments apply the fixed point theory, Green’s functions technique, L1-Carathéodory functions theory, a truncation technique, and Schauder’s fixed point theorem. The technique used consists in application of a Nagumo-type growth condition to nonlinearities and the concept of equiconvergence for recovering the compactness of the associated operators. In the last section, we present an example. PubDate: 2019-01-18 DOI: 10.1007/s10986-019-09419-y
Abstract: Abstract In this paper, we study properties of functions and sequences with a semi-heavy tail, that is, functions and sequences of the form w(x) = e−βxf(x), β > 0, resp., wn = cnfn, 0 < c < 1, where the function f(x), resp., the sequence (fn), is regularly varying. Among others, we give a representation theorem and study convolution properties. The paper includes several examples and applications in probability theory. PubDate: 2018-10-01 DOI: 10.1007/s10986-018-9417-0
Abstract: Abstract In this paper, we consider a risk model by introducing a temporal dependence between the claim numbers under periodic environment, which generalizes several discrete-time risk models. The model proposed is based on the Poisson INAR(1) process with periodic structure. We study the moment-generating function of the aggregate claims. The distribution of the aggregate claims is discussed when the individual claim size is exponentially distributed. PubDate: 2018-10-01 DOI: 10.1007/s10986-018-9412-5
Abstract: Abstract We extend the mapping properties of martingale transforms, decoupling inequalities, differential subordination, and the Stein inequalities to exponential Orlicz spaces. PubDate: 2018-10-01 DOI: 10.1007/s10986-018-9410-7
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