Wirsing translation

E. Wirsing, "Das asymptotische verhalten von summen über multiplikative funktionen. II." Acta Mathematica Academiae Scientiarum Hungaricae Tomus 18 (3-4), 1967, pp. 411-467.

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

$(1.1) \sum_{p\leq x}\lambda(p)\log(p)\sim\tau x \mbox{ (p prime)}$

Determine:

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

($c$ is the Euler-) constant. Special rates are the same type Delange [3]. The same result (1.2) is here under the much weaker assumption

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

However, with the additional. Call $\lambda(p)= O(1)$ and only for $\tau\gt0$ are shown (Theorem 1.1). The terms of $\lambda(p^{v}) (v\geq2)$ are thieves than I, but we want them in the introduction . neglect The same result for complex-function $\lambda$, we get only if $\lambda$ by $|\lambda|$ not significantly different, namely, if $\sum\frac{1}{p}(|\lambda(p)-Re\lambda(p)|)$ converges (Theorem 1.1.1). The special case $\tau=1,|\lambda|\leq1,\sum\frac{1}{p}(1-\lambda(p))$ is convergent was proved by Delange [4] easier to by Rényi [8].

An interesting counterpart to give Erdos and Rényi [7]: convergence $\sum\frac{1}{p}(1-\lambda(p)),\sum\frac{1}{p^{2}}\lambda(p)^{2}$ and $\sum_{p}\sum_{v\geq2}\frac{1}{p^{v}}\lambda(p^{v})$ and for each $\epsilon\gt0$

$\liminf_{x\to\infty}\sum_{x\ltp\leq(1+\epsilon)x}\lambda(p)\frac{\log p}{p}\gt0,$

then (1.2) (with $\tau=1$). Here $\lambda$ will be restricted to the bottom.

If $\lambda^{*}$ is another multiplicative function $|\lambda^{*}|\leq\lambda$, so we could in I with the conditions (1.1) and

$(1.4) \sum_{p\leq x}\lambda^{*}(p)\log p\sim\tau^{*}x$

n is the sum $\sum_{n\leq x}\lambda^{*}(n)$ up to $o(\sum_{n\leq x}\lambda(n))$ identify and, in particular

$\sum_{n\leq x}\lambda^{*}(n)=o(\sum_{n\leq x}\lambda(n))$

show if $\sum_{p}\frac{1}{p}(\lambda(p)-Re\lambda^{*}(p))$ diverges. In the event $\lambda=1$ see Delange [3]. In the present paper we obtain such results without (1.4), some of which (1.1), some of them already with (1.3), if the range of values of $\lambda^{*}$ is suitably restricted. In particular, we prove (Theorem 1.2.2): Do (1.3), $\lambda(p)=O (1)$, $|\lambda^{*}|\leq\lambda$ is $\lambda^{*}$ real-valued, the average of $\lambda^{*}$ exists regarding $\lambda$:

$(1.5) \lim_{x\to\infty}(\sum_{n\leq x}\lambda^{*}(n))(\sum_{n\leq x}\lambda(n))^{-1}$

and has the value

$(1.6) \lim_{x\to\infty}\prod_{p\leq x}\left(1+\frac{\lambda^{*}(p)}{p}+\frac{\lambda^{*}(p^2)}{p^2}+\cdots\right)\left(1+\frac{\lambda(p)}{p}+\frac{\lambda(p^2)}{p^2}+\cdots\right)^{-1}.$

If one specifically for $\lambda$ is the constant 1 and allowed $\lambda^{*}$ only values $\pm1$, then this is the solution of a known problem of Wintner [9] (there) with a supposed proof, see Erdos [6].

For complex-valued functions $\lambda^{*}$, the situation is more complicated. The example $\lambda(n)=1,\lambda^{*}(n)=n^{i}$ shows that the existence of (1.6), the foltg of (1.5) does not if only (1.1), $\lambda(p)=O(1)$ and $|\lambda^{*}|\leq\lambda$ requires. It is then that is $\sum_{n\leq x}\lambda^{*}(n)\sim x^{1+i}(1+i)^{-1}$ while (1.6) has the value 0. But if $|\lambda^{*}|\leq\lambda$ vershärft to the following claim:

$\lambda^{*}(n)=\epsilon(n)\lambda(n),|\epsilon(n)|\leq1.$

There are a number $e^{i\phi}$ of magnitude 1, which is "not an accumulation point of the sequence $(\epsilon(p))$, then (Theorem 1.2.1 follows):

$(1.7) (\sum_{n\leq x}\lambda^{*}(n))(\sum_{n\leq x}\lambda(n))^{-1}\to0\mbox{, resp.}\sim\prod_{p\leq x}\left(1+\frac{\lambda^{*}(p)}{p}+\cdots\right)\left(1+\frac{\lambda(p)}{p}+\cdots\right)^{-1},$

depending on the product does not tend to 0 or.

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