On certain unbounded multiplicative functions in short intervals

Abstract

Recently, Mangerel extended the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions in typical short intervals. In this paper, we combine Mangerel's result with Halász-type result recently established by Granville, Harper and Soundararajan to consider the distribution of a class of multiplicative functions in short intervals. First, we prove cancellation in the sum of the coefficients of the standard L-function of an automorphic irreducible cuspidal representation of \(\mathrm{GL}_m\) over \(\mathbb{Q}\) with unitary central character in typical intervals of length \(h(\log X)^c\) with \(h = h(X) \rightarrow \infty\) and some constant \(c > 0\) (under Vinogradov–Korobov zero-free region and GRC). Then we also establish a non-trivial bound for the product of divisor-bounded multiplicative functions with the Liouville function in arithmetic progressions over typical short intervals.