Theorem 1 Let F:N-->R stand for any function which a) $F$ monotonically weakly increases; b) $F$ tends to infinity; and c) such that $q/F(q)$ tends to infinity. Let Z_F(q) equal the number of divisors of q less than sqrt{ F(q)} minus the number of divisors of q between sqrt{ F(q)} and F(q). Then, on the average, Z_F(q) equals Euler's constant Theorem 2 Fix a in (0,1). Write A for the average number of divisors of n that lie in (0,sqrt{ a n}) minus the number of that lie in (sqrt{ a n},a n)$. Then A= (sum_{ i=1}^{ \lceil { 1-a}/a \rceil} \frac{ 1}{ i}) - ln(1/a).
Categories: math.NT