A problem involving integers all of whose prime divisors belong to given arithmetic progressions

• Published in: Russian mathematical surveys Vol. 71; no. 4; pp. 790 - 792
• Main Author: Changa, M. E.
• Format: Journal Article
• Language: English
• Published: London Mathematical Society, Turpion Ltd and the Russian Academy of Sciences 01.01.2016
• Subjects: Arithmetic; Asymptotic methods; Encounters; Formulas (mathematics); Functions (mathematics); Integers; Mathematical analysis; Progressions
• ISSN: 0036-0279; 1468-4829
• DOI: 10.1070/RM9700

Fix an integer k>2 and r distinct natural numbers l sub(1),...l sub(r)coprime with k and not exceeding k, and assume that r<[phi](k), where [phi](k) is the Euler function. Let M be the set of positive integers such that each prime divisor of each of them is congruent to one of the integers l sub(i)modulo k. For the number of integers in M that do not exceed a given number x, Landau ([1], p. 668) found an asymptotic formula whose leading term has order x(ln x) super()r[phi](k)-1 By using the method of complex integration one can also find asymptotic formulae for the sums of the values of multiplicative functions over the integers in M that do not exceed a given number x. In such problems we often encounter a peculiar phenomenon: the sum of the values of a function over the integers in M can be significantly greater than the analogous sum over the integers subject to no restriction on their prime divisors.