Table of contents

A phenomenon of interdependency between the structure of positive integers and the form of their prime factors is discovered. This paper was prepared for publication by E.A. Karatsuba and M.E. Changa, based on A.A. Karatsuba's drafts and notes from 2007-2008. Details of calculations are due to Changa.

Bibliography: 10 titles.

This paper deals with analytic properties of certain Dirichlet series considered by Davenport and Heilbronn and by Titchmarsh.

Bibliography: 28 titles.

This survey is devoted to the classical problem of flag numbers of convex polytopes, and contains an exposition of results obtained on the basis of connections between the theory of convex polytopes and a number of modern directions of research.

Bibliography: 62 titles.

The main goal of this survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretic problems. Originally designed as a tool for solving these problems, the hypergeometric series have become a connecting link between different areas of number theory and mathematics in general.

Bibliography: 183 titles.

Stones are placed randomly on an board, one at a time, with no more than one stone per unit cell. By means of a two-dimensional cross-classification, a formula is derived for the probability that a row becomes full (with stones) at a time when there is another row that is still entirely empty.

Bibliography: 2 titles.

Let be polynomials in  variables, over the field , and suppose that their degrees are . It was shown by Warning in 1935 that if is the number of common zeros of the polynomials , then . It is the main aim of the present paper to improve on this bound. When the set of common zeros does not form an affine linear subspace in , it is shown for example that if , and that if the are all homogeneous.

Bibliography: 5 titles.