Moment estimates for square-free integers on short intervals

Maria Avdeeva, Princeton University
Fine Hall 601

Square-free integers are known to have asymptotic density 6/(pi^2). Fix some x and let n be distributed uniformly on the integers between 1 and x. Consider the corresponding variance of the number of square-free integers on a short interval [n+1, n+N] and let x tend to infinity. In 1982, R.Hall proved that the limiting variance behaves asymptotically, as N tends to infinity, like C*N^{1/2} for some constant C. In 1987, Hall also derived some estimates for higher moments of this random variable. Following another method, we will obtain a different estimate for the third moment. If time permits, we will also discuss higher moments and generalization of Hall's result to the case of k-free integers.