Volume 234, Nº 5 (2018)

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.

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.

The paper establishes the invariance of the simplex-module algorithm for expanding real numbers α = (α₁, …, α_d) 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 GL_d+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.

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 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.

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

\( {D}_{\rho}\left(x;{\zeta}_4\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_4(n) \)

for an arbitrary fixed ρ > 0. The error term Δ_ρ(x; ζ₄) is defined by

\( {D}_{\rho}\left(x;{\zeta}_4\right)=\frac{\uppi^2{x}^{2+\rho }}{\Gamma \left(\rho +3\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_4(0)+{\Delta}_{\rho}\left(x;{\zeta}_4\right). \)

It is proved that

\( {\Delta}_4\left(x;{\zeta}_4\right)=\Big\{{\displaystyle \begin{array}{ll}O\left({x}^{1/2+\rho +\varepsilon}\right)& \left(1<\rho \le 3/2\right),\\ {}O\left({x}^{9/8+\rho /4}\right)& \left(1/2<\rho \le 1\right),\\ {}O\left({x}^{5/4+\varepsilon}\right)& \left(0<\rho \le 1/2\right)\end{array}} \)

and

\( {\Delta}_{1/2}\left(x;{\zeta}_4\right)=\Omega \left(x{\log}^{1/2}x\right). \)