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Divisor-bounded multiplicative functions in short intervals

Mangerel, Alexander P.

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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.


Mangerel, A. P. (2023). Divisor-bounded multiplicative functions in short intervals. Research in the Mathematical Sciences, 10(12),

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


Published Journal Article (730 Kb)

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