Divisor-bounded multiplicative functions in short intervals

TL;DR

This paper extends the Matomäki-Radziwiłł theorem to a broad class of unbounded multiplicative functions, enabling new estimates of their averages and variances in short intervals, with applications to automorphic forms and divisor functions.

Contribution

It generalizes the Matomäki-Radziwiłł theorem to unbounded multiplicative functions and applies this to automorphic form coefficients and divisor functions in short intervals.

Findings

Classical asymptotics for Fourier coefficients persist in short intervals.

Variance estimates for moments of automorphic form coefficients are established.

Hooley's Δ-function has average value growing with log log X in short intervals.

Abstract

We extend the Matom\"{a}ki-Radziwi\l\l{} 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 $c = c_f \geq 0$ 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 $h = h(X) \rightarrow \infty$. We also generalize this result to sequences $\{|\lambda_{\pi}(n)|^2\}_n$, where $\lambda_{\pi}(n)$ is the $n$th coefficient of the standard $L$-function of an automorphic representation $\pi$ with unitary central character for $GL_m$, $m \geq 2$, 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.