Divisor-bounded multiplicative functions in short intervals

Abstract

AbstractWe 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 $$h(\log X)^c$$ h ( log X ) c , with $$h = h(X) \rightarrow \infty $$ h = h ( X ) → ∞ and where $$c = c_f \ge 0$$ c = c f ≥ 0 is determined by the distribution of $$\{|f(p) |\}_p$$ { | 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 $$|\lambda _f(n) |^2$$ | λ f ( n ) | 2 , where $$\{\lambda _f(n)\}_n$$ { λ 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 $$h\log X$$ h log X , if $$h = h(X) \rightarrow \infty $$ h = h ( X ) → ∞ . We also generalize this result to sequences $$\{|\lambda _{\pi }(n) |^2\}_n$$ { | λ π ( n ) | 2 } n , where $$\lambda _{\pi }(n)$$ λ π ( n ) is the nth coefficient of the standard L-function of an automorphic representation $$\pi $$ π with unitary central character for $$GL_m$$ G L m , $$m \ge 2$$ m ≥ 2 , provided $$\pi $$ π 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 $$\{|\lambda _f(n) |^{\alpha }\}_n$$ { | λ f ( n ) | α } n over intervals of length $$h(\log X)^{c_{\alpha }}$$ h ( log X ) c α , with $$c_{\alpha } > 0$$ c α > 0 explicit, for any $$\alpha > 0$$ α > 0 , as $$h = h(X) \rightarrow \infty $$ h = h ( X ) → ∞ . Finally, we show that the (non-multiplicative) Hooley $$\Delta $$ Δ -function has average value $$\gg \log \log X$$ ≫ log log X in typical short intervals of length $$(\log X)^{1/2+\eta }$$ ( log X ) 1 / 2 + η , where $$\eta >0$$ η > 0 is fixed.