Roughly speaking, the spectrum of multiplicative functions is the set of all possible mean values. In this paper, we are interested in the spectra of multiplicative functions supported over powerfull numbers. We prove that its real logarithmic spectrum takes values from to 1 while it is known that the logarithmic spectrum of real multiplicative functions over all natural numbers takes values from 0 to 1. In the course of this study, we correct and complete the proof of Granville and Soundararajan on the spectrum of all multiplicative functions.

中文翻译：

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所有乘法函数的谱及其对强大数字的应用

粗略地说，乘法函数的谱是所有可能平均值的集合。在本文中，我们感兴趣的是在强大的数字上支持的乘法函数的谱。我们证明了它的实对数谱的取值是从 到 1，而已知所有自然数的实乘函数的对数谱的取值是从 0 到 1。在本研究过程中，我们纠正并完成了 Granville 的证明和 Soundararajan 在所有乘法函数的谱上。