The spectrum of multiplicative functions
Andrew Granville, K. Soundararajan
Abstract
Let S be a subset of the unit disk, and let F(s) denote the class of completely multiplicative functions f such that f(p) is in S for all primes p. The authors' main concern is which numbers arise as mean-values of functions in F(s). More precisely, let GammaN(S) = 1/N sumn <= N f(n): f in F(S) and Gamma(S) = limN -> infinity GammaN(s). The authors call Gamma(S) the spectrum of the set S, and study its properties.
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