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Roughly speaking, the spectrum of multiplicative functions is the set of allpossible mean values. In this paper, we are interested in the spectra ofmultiplicative functions supported over powerful numbers. We prove that itsreal logarithmic spectrum takes values from $- \sqrt{2} / (4 + \sqrt{2}) =-0.26160...$ to $1$ while it is known that the logarithmic spectrum of realmultiplicative functions over all natural numbers takes values from $0$ to $1$.In the course of this study, we correct a proof of Granville and Soundararajanconcerning contribution of small primes in the study of mean value ofmultiplicative functions.

Spectrum of multiplicative functions over powerful numbers