6 research outputs found
On consecutive values of random completely multiplicative functions
In this article, we study the behavior of consecutive values of random
completely multiplicative functions whose values are i.i.d.
at primes. We prove that for uniform on the unit circle, or uniform on
the set of roots of unity of a given order, and for fixed , are independent if is large enough. Moreover, with the same
assumption, we prove the almost sure convergence of the empirical measure
when goes to
infinity, with an estimate of the rate of convergence. At the end of the paper,
we also show that for any probability distribution on the unit circle followed
by , the empirical measure converges almost surely when