DOIAbstract
AbstractLet 1=d1(n)<d2(n)<⋯<dτ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max1⩽i<τ(n)di+1(n)/di(n)⩽y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let u=logx/logy, and v=logx/logz. We show that x−1D(x,y,z)logz is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)≍(1−u/v)/(u+1) for v⩾3+ε. Finally, we show that d(u,v)=e−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x−1D(x,y,1), for fixed u. This leads to a new estimate for d(u)
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