tailieunhanh - Báo cáo toán học: "On a Multiplicative Partition Function"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: On a Multiplicative Partition Function. | On a Multiplicative Partition Function Yifan Yang Department of Mathematics University of Illinois Submitted October 5 2000 Accepted April 12 2000 MR Subject Classification primary 11N60 secondary 05A18 11P82 Abstract Let D s V1 1 amm-s be the Dirichlet series generated by the infinite product m 2 1 k s . The value of am for squarefree integers m with n prime factors depends only on the number n and we let f n denote this value. We prove an asymptotic estimate for f n which allows us to solve several problems raised in a recent paper by M. V. Subbarao and A. Verma. 1 Introduction and Statements of Results Let D s V1 1 amm s be the Dirichlet series generated by the infinite product m 2 1 - k- s . The coefficients am denote the excess of the number of unordered representations of m as a product of an even number of distinct integers 1 over the number of representation of m as a product of an odd number of distinct integers 1. The Dirichlet series D s is closely related to the generating Dirichlet series in the Fac-torisatio Numerorum problem of Oppenheim see 6 . Indeed if we let bm denote the number of unordered representations of m as a product of integers 1 not necessarily distinct then we have Ym 1 bmm-s D s -1. Thus by the Mobius inversion formula the numbers am and bm are related by the identity am Y dimụ d bm d. Oppenheim 6 showed that x XXbm 2p logx 3 4 In 3 E. R. Canfield P. Erdos and C. Pomerance proved that if m is an integer such that bn bm for all n m then bm m exp 1 o 1 logm log3 m log2 mg where logfc denotes the k-times iterated logarithm. In this paper we consider the more difficult problem of investigating the asymptotic behavior of the numbers am. This problem was raised by M. V. Subbarao who observed THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R19 1 that am 0 1 for all positive integers m with at most four prime factors and asked whether this is true for all m. It is easy to see that for a positive integer m 1 the coefficient am depends only on .