卷 225, 编号 6 (2017)

An automorphic structure on a Lie group consists of the Hecke–Shimura ring of an arithmetic discrete subgroup and a linear representation of the ring on an invariant space of automorphic forms given by Hecke operators. The paper is devoted to interactions (transfer homomorphisms) of the Hecke–Shimura rings of integral symplectic groups and integral orthogonal groups of integral positive definite quadratic forms. Bibliography: 10 titles.

A number of general results concerning the strong form of asymptotic formulas of Voronovskaya–Bernstein type is established. Several examples of their application to some approximation methods are provided.

In this paper, the capacity and module of a condenser, and also some functional classes on a sub-Finsler space are defined. Their general properties are studied, and the equality of the capacity and module of a condenser is proved.

The simplex-module algorithm for expansion of algebraic numbers α = (α₁, . . . , α_d) in multidimensional continued fractions is suggested. The method is based on minimal rational simplices s, where α ∈ s, and Pisot matrices P_α, for which \( \widehat{\alpha} \) = (α₁, . . . , α_d, 1) is an eigenvector. A multidimensional generalization of the Lagrange theorem is proved.

For p > 1, the notion of the p-harmonic Robin radius of a domain in the space ℝⁿ, n ≥ 2, is introduced. In the case where the corresponding part of the boundary degenerates, the Robin–Neumann radius is considered. The monotonicity of the p-harmonic Robin radius under some deformations of a domain is proved. Some extremal decomposition problems in the Euclidean space are solved. The definitions and proofs are based on the technique of moduli of curve families. Bibliography: 23 titles.

The zeta function associated with the category of finite Abelian groups is defined and studied. This zeta function is expressed in terms of the Riemann zeta function.

Let Pk(n) be the difference between the number of points of the integer lattice contained in the ball \( {y}_1^2+\cdots {y}_k^2\le n \) and the volume of this ball. The paper investigates the asymptotic behavior of

the sums

\( \sum_{n\le x}{P}_k(n)\kern0.5em \left(k\ge 4\right),\kern0.5em \sum_{n\le x}{P}_3^2(n),\kern0.5em \sum_{n\le x}{P}_4^2(n)\kern1em as\kern0.5em x\to +\infty . \)