The Tauberian theorems for the slowly variating with residual functions and their applications

E. Wirsing setted up a problem in 1967 year: Is it possible to reduce the estimation $\sum\limits_{n\le x}f(n)=o\left(\frac {x}{\log x}\sum\limits_{n\le x}\frac{f(n)}{n}\right), x\to \infty(1)$ from the estimation $\sum\limits_{p\le x}\frac {f(p)\log p}{p}=o(\log x),x\to \infty(2)$. Here $n$ is a positive enteger, $p$ is a prime number. Let us denote the right-side sum in formula (2) by $m(x)$. B. V. Levin and A. S. Finelabe had proved that the statement (2) did not emply the statement (1). The function $f(n)$ of their conterexample is such that $m(x)$ is bounded. But if $m(x)$ is not bounded that Wirsing problem is opened. Two the Tauberian theorems is proved in this paper and it is established that if $m(x)$ is not bounded that the condition (2) is equivalent that $m(e^{t})$ is slowly variating with the residual.