Divisor-bounded multiplicative functions in short intervals

Mangerel, Alexander P.

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(logX)c , with h=h(X)→∞ and where c=cf≥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 |λf(n)|2 , where {λf(n)}n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length hlogX , if h=h(X)→∞ . We also generalize this result to sequences {|λπ(n)|2}n , where λπ(n) is the nth coefficient of the standard L-function of an automorphic representation π with unitary central character for GLm , m≥2 , provided 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 {|λf(n)|α}n over intervals of length h(logX)cα , with cα>0 explicit, for any α>0 , as h=h(X)→∞ . Finally, we show that the (non-multiplicative) Hooley Δ -function has average value ≫loglogX in typical short intervals of length (logX)1/2+η , where η>0 is fixed.

Citation

Mangerel, A. P. (2023). Divisor-bounded multiplicative functions in short intervals. Research in the Mathematical Sciences, 10(12), https://doi.org/10.1007/s40687-023-00376-0

• Journal Article Type: Article
• Acceptance Date: Jan 16, 2023
• Online Publication Date: Feb 18, 2023
• Publication Date: 2023
• Deposit Date: Feb 22, 2023
• Publicly Available Date: Feb 22, 2023
• Journal: Research in the Mathematical Sciences
• Print ISSN: 2522-0144
• Electronic ISSN: 2197-9847
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 10
• Issue: 12
• DOI: https://doi.org/10.1007/s40687-023-00376-0