Let $a(n)$ be an arithmetic function such that $$ \sum_{n=1}^{\infty}a(n)/n^s=f(s)łog g(s)+h(s), $$ where $f(s)$ is analytic for Re $(s)>1/2$ and bounded for Re $(s)\ge 1/2+\varepsilon$, $g(s)$ is a zeta-like function, $h(s)$ is analytic and bounded for Re $(s)\ge 1/2+\varepsilon$. Then $$ \sum_{nłe x}a(n)=xłeft[b_1/łog x+\cdots+b_m/łog^mx+O(1/łog^{m+1}x)\right] $$ with arbitrary fixed $m\ge 1$, $b_1=f(1)$ and computable constants $b_2,\cdots,b_m$.