On the distribution mod 1 of $\alpha \sigma(n)$

Abstract

The sequence xn = F(n) + ®¾(n) (mod 1) is investigated, where ¾(n) = sum of divisors of n, F is an additive arithmetical function. In an earlier paper De Koninck and the author proved that xn mod 1 is uniformly distributed if the approximation type of ® is ¯nite, and formulated the conjecture that it holds for every irrational ®. In this paper it is proved that the conjecture is not true in general, and it is true if ® 2 K¤. K¤ is de¯ned as follows. Let Mx = Q p prp , p runs over the primes and rp is the integer part of the number stated in the right hand side of (2.7). Let K = Kx be the set of those irrational ®, for which minHjMx kH®kx > 1 holds for every large x, K¤ = f® j j® 2 Kg for every j = 1; 2; : : : .