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On large deviations of additive functions

Radziwill, Maksym (2009) On large deviations of additive functions. . (Unpublished)

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We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. The Kac-Kubilius model suggests that the distribution of values of a given additive function can be modeled by a sum of random variables. We show that the model is accurate (in a large deviation sense) when one is looking at values of the additive function around its mean, but fails, by a constant multiple, for large values of the additive function. We believe that this phenomenon arises, because the model breaks down for the values of the additive function on the "large" primes. In the second part of the paper, we are motivated by a question of Elliott, to understand how much the distribution of values of the additive function on primes determines, and is determined by, the distribution of values of the additive function on all of the integers. For example, our main theorem, implies that a positive, strongly additive function is roughly Poisson distributed on the integers if and only if it is 1+o(1) or o(1) on almost all primes.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Additional Information:The author would like to acknowledge financial support from Université de Montréal (summer 2008), McGill University (summer 2009) and NSERC.
Funding AgencyGrant Number
Université de MontréalUNSPECIFIED
McGill UniversityUNSPECIFIED
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Classification Code:2000 Mathematics Subject Classification. Primary: 11N64 Secondary: 11N60, 11K65, 60F10
Record Number:CaltechAUTHORS:20180614-141222705
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87123
Deposited By: George Porter
Deposited On:14 Jun 2018 22:42
Last Modified:03 Oct 2019 19:52

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