Vol. 26, No. 1, 1968

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Some continuity properties of the Schnirelmann density

R. L. Duncan

Vol. 26 (1968), No. 1, 57–58

Let S denote the set of all infinite increasing sequences of positive integers. For all A = {an} and B = {bn} in S, define the metric ρ(A,B) = 0 if A = B, i.e., if an = bn for all n and ρ(A,B) = 1∕k otherwise, where k is the smallest value of n for which anbn. Similar metrics have been considered previously [1, 21.

Our purpose here is to discuss several continuity properties of the Schnirelmann density d(A) = inf A(n)∕n, where A(n) is the number of elements of A not exceeding n. In particular, we obtain a characterization of the set of all sequences having density zero as the set of points of continuity of d(A).

Mathematical Subject Classification
Primary: 10.49
Received: 18 July 1967
Published: 1 July 1968
R. L. Duncan