An estimate for multiplicative function

Given a multiplicative function $f$ with divisor bound $|f|\le \tau_k$, where $k$ is a nonnegative real number. We consider the Dirichlet series $$ F(s)=\sum_{n=1}^\infty \dfrac{f(n)}{n^s}. $$ Since $$ \sum_{n=1}^\infty \dfrac{\tau_k(s)}{n^s}=\zeta(s)^k, $$ $F(s)$ absolutely converges on the region $\sigma>1$. Now we proceed by assuming an average property for $f$. For example, we may assume there exists a complex number, say $a$, such that for all $x\ge 2$ we have $$\sum_{n\le x} \dfrac{f(p)}{p}\log p=a\log x+R$$ where the remainder $R$ has a asymptotic behaviour of class $\mathcal{O}(1)$ , $\mathcal{O}_A\Big(\frac{x}{(\log x)^A} \Big)$ and so on. I want to get an asymptotic formula for $F(0)$ with the following structure $$ \sum_{n\le x}\dfrac{f(n)}{n}= \dfrac{\mathfrak{S}f}{\Gamma(a+1)}(\log x)^a +\mathcal{O}_{k}\left((\log x)^{\text{Re}a-1}\right) $$ where $$ \mathfrak{S}f=\prod_{p}\left(1-\dfrac{1}{p}\right)^a\left(1+\dfrac{f(p)}{p}+\dfrac{f(p^2)}{p^2}+\cdots\right). $$ The most important thing is the remainder of the asymptotic formula is $(\log x)^{\text{Re}a-1}$. $(\log x)^{|a|-1},(\log x)^{\text{Re} a-c}$ are not good enough. What average propety can deduce this kind of asymptotic formula?