Том 234, № 5 (2018)
- Жылы: 2018
- Мақалалар: 12
- URL: https://journals.rcsi.science/1072-3374/issue/view/14964
Article
In Memory of Galina Vasil’evna Kuz’mina (1929–2017)
Alternating Sums of Elements of Continued Fractions and the Minkowski Question Mark Function
Аннотация
The paper considers the function A(t) (0 ≤ t ≤ 1), related to the distribution of alternating sums of elements of continued fractions. The function A(t) possesses many properties similar to those of the Minkowski function ?(t). In particular, A(t) is continuous, satisfies similar functional equations, and A′(t) = 0 almost everywhere with respect to the Lebesgue measure. However, unlike ?(t), the function A(t) is not monotonically increasing. Moreover, on any subinterval of [1, 0], it has a sharp extremum.
Modules of Families of Vector Measures on a Riemann Surface
Аннотация
The paper considers a Riemann surface (in a broad sense of the term in the Hurwitz–Courant terminology) and an open set with a compact closure on this surface. It is proved that, following Aikawa–Ohtsuka, with a condenser on a given open set one can associate a family of vector measures, whose modules are computed directly with the aid of the weighted capacity (with Muchenhoupt weight) of the condenser.
Linear-Fractional Invariance of Multidimensional Continued Fractions
Аннотация
The invariance of the simplex-karyon algorithm for expanding real numbers α = (α1, …, αd) in multidimensional continued fractions under linear-fractional transformations \( {\alpha}^{\prime }=\left({\alpha}_1^{\prime },\dots, {\alpha}_d^{\prime}\right)=U\left\langle \alpha \right\rangle \) with matrices U from the unimodular group GLd+1(ℤ) is established. For the transformed collections α′, convergents of the best approximations are found.
Linear-Fractional Invariance of the Simplex-Module Algorithm for Expanding Algebraic Numbers in Multidimensional Continued Fractions
Аннотация
The paper establishes the invariance of the simplex-module algorithm for expanding real numbers α = (α1, …, αd) in multidimensional continued fractions under linear-fractional transformations \( {\alpha}^{\prime }=\left({\alpha}_1^{\prime },\dots, {\alpha}_d^1\right)=U\left\langle \alpha \right\rangle \) with matrices U from the unimodular group GLd+1(ℤ). It is shown that the convergents of the transformed collections of numbers α′ satisfy the same recurrence relation and have the same approximation order.
Localized Pisot Matrices and Joint Approximations of Algebraic Numbers
Аннотация
A development of the simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions is proposed. To this end, localized Pisot matrices are constructed, whose eigenvalues with moduli less than one are contained in an interval of small length. Such Pisot matrices generate continued fractions whose convergents are arbitrarily close to the best approximations.
An Inverse Factorial Series for a General Gamma Ratio and Related Properties of the Nørlund–Bernoulli Polynomials
Аннотация
The inverse factorial series expansion for the ratio of products of gamma functions whose arguments are linear functions of the variable is found. A recurrence relation for the coefficients in terms of the Nørlund–Bernoulli polynomials is provided, and the half-plane of convergence is determined. The results obtained naturally supplement a number of previous investigations of the gamma ratios, which began in the 1930-ies. The expansion obtained in this paper plays a crucial role in the study of the behavior of the delta-neutral Fox’s H-function in the neighborhood of its finite singular point. A particular case of the inverse factorial series expansion is used in deriving a possibly new identity for the Nørlund–Bernoulli polynomials.
On Cubic Exponential Sums and Gauss Sums
Аннотация
Let eq be a nontrivial additive character of a finite field ????q of order q ≡ 1(mod 3) and let ψ be a cubic multiplicative character of ????q, ψ(0) = 0. Consider the cubic Gauss sum and the cubic exponential sum
It is proved that for all nonzero a, b ∈ ????q,
where the summation runs over all nonzero n ∈ ????q.
Weighted Modules and Capacities on a Riemann Surface
Аннотация
On a Riemann surface (in the broad sense of the word in the terminology of Hurwitz–Courant), the weighted capacity and module (with a Muckenhoupt weight) of a condenser with a finite number of plates are defined. The equality of the capacity and module of a condenser is proved, which solves a Dubinin problem.
On Riesz Means of the Coefficients of Epstein’s Zeta Functions
Аннотация
Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius \( \sqrt{n} \). The generating function
is Epstein’s zeta function. The paper considers the Riesz mean of the type
where ρ > 0; the error term Δρ(x; ζ3) is defined by
K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that
In the present paper, it is proved that
and the Riesz means of the coefficients of ζk(s), k ≥ 4, are studied.
Lattice Points in the Four-Dimensional Ball
Аннотация
Let r4(n) denote the number of representations of n as a sum of four squares. The generating function ζ4(s) is Epstein’s zeta function. The paper considers the Riesz mean
for an arbitrary fixed ρ > 0. The error term Δρ(x; ζ4) is defined by
It is proved that
and