Divisor-bounded multiplicative functions in short intervals

We extend the Matomaki-Radziwill 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 fin typical intervals of length h(log X)(C), with h = h(X)-> infinity and where c = c(f) >= 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 h log X, if h = h(X)-> infinity. 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 >= 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)-> infinity. Finally, we show that the (non-multiplicative) Hooley delta-function has average value >> log log X in typical short intervals of length (log X)(1/2+eta), where eta > 0 is fixed.