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 primes. Our result allows us estimate averages such function f in typical intervals length $$h(\log X)^c$$ h ( log X ) c , with $$h = h(X) \rightarrow \infty $$ = → ∞ and where $$c c_f \ge 0$$ f ≥ 0 is determined distribution $$\{|f(p) |\}_p$$ { | p } an explicit way. give three applications. First, we show classical Rankin–Selberg-type asymptotic formula for partial sums $$|\lambda _f(n) |^2$$ λ n 2 $$\{\lambda _f(n)\}_n$$ sequence normalized Fourier coefficients primitive non-CM holomorphic cusp form, persists short $$h\log X$$ if . also generalize this sequences $$\{|\lambda _{\pi }(n) |^2\}_n$$ π $$\lambda }(n)$$ n th coefficient standard L -function automorphic representation $$\pi unitary central character $$GL_m$$ G L m $$m 2$$ provided satisfies generalized Ramanujan conjecture. Second, using recent developments theory forms variance all positive real moments |^{\alpha }\}_n$$ α over X)^{c_{\alpha }}$$ $$c_{\alpha } > > explicit, any $$\alpha as Finally, (non-multiplicative) Hooley $$\Delta Δ has average value $$\gg \log ≫ $$(\log X)^{1/2+\eta }$$ 1 / + η $$\eta >0$$ fixed.