The Möbius and Liouville functions in all short intervals of length x^θ, θ > 69/127

We show that for every fixed θ > 69/127 = 0.5433… and ε > 0, Σ_{x 0.55. We follow their argument, which uses Ramaré's identity to extract a small prime factor, and replace the Heath-Brown–Iwaniec type I/II lemma used there by a trilinear mean value estimate for the product of a partial sum of ζ with two arbitrary Dirichlet polynomials. This estimate is proved with level sets, a Kusmin–Landau bound at low frequencies, the mean value theorem, the fourth moment of partial sums of ζ, van der Corput exponent pairs, and the large value estimates of Guth and Maynard. Their Theorem 1.1 alone gives θ > 41/75 = 0.5466…; adding their Proposition 12.1 gives 69/127. The exponent inequality required over all configurations is verified by computer with exact rational certificates. The same argument, with one further change, extends to the same ranges the asymptotic formula of Matomäki and Teräväinen for the number of products of two primes in all short intervals. The asymptotic count is the part that may constitute a new contribution there: the existence of products of two primes in all intervals of length x^θ, θ > 0.525, already follows from results on primes. The improvement is small, and it was anticipated by Matomäki and Teräväinen and by Guth and Maynard."