Vol. 34, No. 2, 1970

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A density which counts multiplicity

Robert E. Dressler

Vol. 34 (1970), No. 2, 371–378

P. Erdös, using analytic theorems, has proven the following results: Let f(x) be the number of integers m such that ϕ(m) x, where ϕ is the Euler function, and let g(x) be the number of integers n such that σ(n) x, where σ is the usual sum of divisors function. Then there are positive (but undetermined) constants 01 and c2 such that f(x) = c1x + o(x) and g(x) = c2(x) + o(x). The constants c1 and c2 can be calculated using complex analysis including the Wiener-Ikehara Theorem. A major purpose of this paper is to give an elementary proof that limx→∞f(x)∕x exists and, in the process, calculate the value of the limit. These considerations of multiplicity motivate a generalization of natural density which counts multiplicity. This paper contains an investigation of this generalization.

Mathematical Subject Classification
Primary: 10.50
Received: 13 May 1969
Published: 1 August 1970
Robert E. Dressler