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 $latex \sum_ (n \leq x) \lambda (n)$ for nonnegative multiplicative functions $latex \lambda$ essentially under the condition

$latex (1.1) \sum_{p\leq x}\lambda(p)\log(p)\tilde\tau x (p prime)$

Determine

$latex (1.2) \sum_{n\leq x}\lambda(n)\tilde\frac{e^{-ct}}{\Gamma(\tau)}\frac{x}{\log x}\prod_{p\leq x}\left(1+\frac{\lambda(p)}{p}+\frac{\lambda(p^{2})}{p^{2}}+\cdots\right)$

($latex c$ is the Euler-) constant. Special rates are the same type Delange [3]. The same result

(1.2) is here under the much weaker assumption

$latex (1.3) \sum_{p\leq x}\lambda(p)\frac{\log p}{p}\tilde\tau\log x$

However, with the additional. Call $latex \lambda(p)= O(1)$ and only for tau \> are shown 0 (Theorem 1.1). The terms of $latex \lambda(p^{v}) (v\geq2)$ are thieves than I, but we want them in the introduction . neglect The same result for complex-function $latex \lambda$, we get only if $latex \lambda$ by $latex |\lambda|$ nich significantly different, namely, if

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