1. Introduction. It is the object of this paper to investigate the function γ(m), the number of representations of m in the form
where . It is shown that γ(m) is always equal to the number of odd divisors of m, so that for example γ(2 k ) = 1, this representation being the number 2 k itself. From this relationship the average order of γ(m) is deduced ; this result is given in Theorem 2. By a method due to Kac , it is shown in §3 that the number of positive integers for which γ(m) does not exceed a rather complicated function of n and ω, a real parameter, is asymptotically nD(ω), where D(ω) is the probability integral
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