Improved bounds for the two-point Chowla conjecture at almost all scales
The Liouville function λ(n) is defined to be +1 if n is a product of an even number of primes, and -1 otherwise. In many ways, the Liouville function is expected to behave like a random sequence of +1’s and -1’s. In- stances of this phenomenon are intimately connected with the distribution of primes. For example, the statement that n≤x λ(n) = Oε(x1/2+ε) is equivalent to the Riemann hypothesis. The two-point Chowla conjecture predicts that the average of λ(n)λ(n+1) over n ≤ x tends to zero as x → ∞. I will discuss quantitative bounds for a version of this problem at almost all scales x, improving a result of Helfgott and Radziwill.