Improved bounds for multiplicative functions in almost all short intervals

We refine the Matomäki-Radziwiłł method for short averages of multiplicative functions.

For the Liouville function and multiplicative functions supported on smooth numbers, we prove decay bounds that are essentially optimal with the Matomäki-Radziwiłł method.

The key new ingredient is a sharper treatment of the sieve error, achieved by introducing a more widely separated final prime range when restricting to integers with typical factorizations.

We additionally give a weaker but still improved bound for arbitrary 1-bounded multiplicative functions and discuss some limitations of the method.

As an application, we give an improved bound for the averaged Chowla conjecture of Matomäki-Radziwiłł-Tao that seems essentially best possible via their method.