A closer look at some new identities

Geoffrey B Campbell

doi:10.1155/s0161171298000805

Abstract

We obtain infinite products related to the concept of visible from the origin point vectors. Among these is urn:x-wiley:01611712:media:ijmm289376:ijmm289376-math-0001 in which φ3(k) denotes for fixed k, the number of positive integer solutions of (a, b, k) = 1 where a < b < k, assuming (a, b, k) is the gcd(a, b, k).

Highlights

In several of the author’s recent papers a class of new elementary infinite product identities was introduced. These were gven the name vpv identities, meaning visible pom/vec/or tdenttttes, due to the fact that they involved summation over so called visible lattice points in various dimensional spaces For an introduction to the idea of visible lattice points see Apostol [2], where their distribution Is calculated The identities which occurred most frequently in Campbell [7] were often related to each other by vpv lattice sums dividing up space into radial regions from the origin These identities when grouped were called compamon tden/ttles, because of their interdependence We used the operators defined by
[6] Paper [6] contained results on Dirichlet summations such as those connected with Ramanujan arithmetical nctions, and was related to a method of Meinardus for obtaining asymptotic estimates of coeNcients from infinite products In [7], some her identities denttes of ths type were given and the existence of yet others was indicated In the present note we gve results derived from considerations similar to those in
= The dentities of sectmn have interesting cases in which variables other than approach unity given certain modificauons In Campbell [5] we find the identity where p is the Euler totient function

Summary

Introduction

In several of the author’s recent papers (see Campbell [5,6,7,8,9]) a class of new elementary infinite product identities was introduced. Some of the methods of these papers are applicable to vpv sums and products In Campbell [5,6,7,8,9] the following companion identities were given.

Results

Conclusion