Divisor-bounded multiplicative functions in short intervals

Abstract

We extend the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typical intervals of length h(log X)^c, with h = h(X) rightarrow infty and where c = c_f ge 0 is determined by the distribution of {|f(p) |}_p in an explicit way. We give three applications. First, we show that the classical Rankin–Selberg-type asymptotic formula for partial sums of |lambda _f(n) |^2, where {lambda _f(n)}_n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length hlog X, if h = h(X) rightarrow infty . We also generalize this result to sequences {|lambda _{pi }(n) |^2}_n, where lambda _{pi }(n) is the nth coefficient of the standard L-function of an automorphic representation pi with unitary central character for GL_m, m ge 2, provided pi satisfies the generalized Ramanujan conjecture. Second, using recent developments in the theory of automorphic forms we estimate the variance of averages of all positive real moments {|lambda _f(n) |^{alpha }}_n over intervals of length h(log X)^{c_{alpha }}, with c_{alpha } > 0 explicit, for any alpha > 0, as h = h(X) rightarrow infty . Finally, we show that the (non-multiplicative) Hooley Delta -function has average value gg log log X in typical short intervals of length (log X)^{1/2+eta }, where eta >0 is fixed.