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Published June 14, 2018 | Submitted
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A converse to Halász's theorem


We show that the distribution of large values of an additive function on the integers, and the distribution of values of the additive function on the primes are related to each other via a Levy Process. As a consequence we obtain a converse to an old theorem of Halasz. Halasz proved that if f is an strongly additive function with f (p) ∈ {0, 1}, then f is Poisson distributed on the integers. We prove, conversely, that if f is Poisson distributed on the integers then for most primes p, f(p) = o(1) or f(p) = 1 + o(1).

Additional Information

The author is partially supported by a NSERC PGS-D award. This is part of author's undergraduate thesis, written under the direction of Andrew Granville. The author would like to thank first and foremost Andrew Granville. There is too much to thank for, so it is simpler to note that this project would not surface without his constant support. Also, the author would like to thank Philippe Sosoe for proof-reading a substantial part of the old manuscript of this paper.

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