On the distribution of k-th power free integers, II

The indicator function of the set of k-th power free integers is naturally extended to a random variable X^(k)({dot operator}) on (ℤ○,λ), where ℤ○ is the ring of finite integral adeles and λ is the Haar probability measure. In the previous paper, the first author noted the strong law of large numbers for {X^(k)({dot operator}+n)}^∞_n=1, and showed the asymptotics: E^λ[(Y^(k)_N)²]{equivalent to}1 as N→∞, where Y^(k)_N(x):=N^-1/2k∑^N_n=1(X(k)(x+n)-1/ζ(k)). In the present paper, we prove the convergence of E^λ[(Y^(k)_N)²]. For this, we present a general proposition of analytic number theory, and give a proof to this.