Wirsing translation

E. Wirsing, "Das asymptotische verhalten von summen über multiplikative funktionen. II." Acta Mathematica Academiae Scientiarum Hungaricae Tomus 18 (3-4), 1978, pp. 411-467.

English Translation by: Google Translator

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In I we have the asymptotic behavior of the sum [math]\displaystyle{ \sum_{n \leq x} \lambda (n) }[/math] for nonnegative multiplicative functions [math]\displaystyle{ \lambda }[/math] essentially under the condition

[math]\displaystyle{ (1.1) \sum_{p\leq x}\lambda(p)\log(p)\sim\tau x \mbox{ (p prime)} }[/math]

Determine

[math]\displaystyle{ (1.2) \sum_{n\leq x}\lambda(n)\sim\frac{e^{-ct}}{\Gamma(\tau)}\frac{x}{\log x}\prod_{p\leq x}\left(1+\frac{\lambda(p)}{p}+\frac{\lambda(p^{2})}{p^{2}}+\cdots\right) }[/math]

([math]\displaystyle{ c }[/math] is the Euler-) constant. Special rates are the same type Delange [3]. The same result (1.2) is here under the much weaker assumption

[math]\displaystyle{ (1.3) \sum_{p\leq x}\lambda(p)\frac{\log p}{p}\sim\tau\log x }[/math]

However, with the additional. Call [math]\displaystyle{ \lambda(p)= O(1) }[/math] and only for [math]\displaystyle{ \tau\gt 0 }[/math] are shown (Theorem 1.1). The terms of [math]\displaystyle{ \lambda(p^{v}) (v\geq2) }[/math] are thieves than I, but we want them in the introduction . neglect The same result for complex-function [math]\displaystyle{ \lambda }[/math], we get only if [math]\displaystyle{ \lambda }[/math] by [math]\displaystyle{ |\lambda| }[/math] not significantly different, namely, if

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