Integers with A
نویسنده
چکیده
In this note we prove only the important special case (1.1), omitting the parts of the argument required for other cases. In addition, we present an alternate proof, dating from 2002, of the lower bound implicit in (1.1). This proof avoids the use of results about uniform order statistics required in [3], and instead utilizes the cycle lemma from combinatorics. Although shorter and technically simpler than the argument in [3], this method is not useful for counting integers with a prescribed number of divisors in (y, 2y]. We mention here one of the applications of (1.1), a 50-year old problem of Erdős ([1], [2]) known colloquially as the “multiplication table problem”. Let A(x) be the number of positive integers n ≤ x which can be written as n = m1m2 with each mi ≤ √ x. Then
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