Abstract
We determine the order of magnitude of up to factors of size , where is a Steinhaus or Rademacher random multiplicative function, for all real .
In the Steinhaus case, we show that on this whole range. In the Rademacher case, we find a transition in the behavior of the moments when , where the size starts to be dominated by “orthogonal” rather than “unitary” behavior. We also deduce some consequences for the large deviations of .
The proofs use various tools, including hypercontractive inequalities, to connect with the -th moment of an Euler product integral. When is large, it is then fairly easy to analyze this integral. When is close to 1 the analysis seems to require subtler arguments, including Doob’s maximal inequality for martingales.
Citation
Adam J Harper. "Moments of random multiplicative functions, II: High moments." Algebra Number Theory 13 (10) 2277 - 2321, 2019. https://doi.org/10.2140/ant.2019.13.2277
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