Abstract:

Let Xn, n  1, be a sequence of independent random variables satisfying
P(Xn = 0) = 1 − P(Xn = an) = 1 − 1/pn, where an, n  1, is a sequence of real
numbers, and pn is the nth prime,set FN(x) = P

NP
n=1
Xn  x

. The authors investigate a
conjecture of Erd¨os in probabilistic number theory and show that in order for the sequence
FN to be weakly convergent, it is both sufficient and necessary that there exist three
numbers x0 and x1 < x2 such that limsup
N!1
(FN(x2) − FN(x1)) > 0 holds, and L0 =
lim
N!1
FN(x0) exists. Moreover, the authors point out that they can also obtain the same
result in the weakened case of liminf
n!1
P(Xn = 0) > 0.

Key words: Erd¨os&rsquo, conjecture, additive arithmetic function, sums of independent random
variables, essential convergence, weak convergence