We denote $\log_2$ the logarithm function in base 2, defined by $\log_2(x) = \frac{\ln(x)}{\ln(2)}$ for all real $x > 0$. With $w = \frac{1}{\lambda-1}$ and $\mathbf{f} = (1+w)\delta - w\mathbf{1}$, show that, for $s$ real sufficiently large, $$\frac{1}{L_{\mathbf{f}}(s)} = 1 + \sum_{m=2}^{\infty} m^{-s} \sum_{k=1}^{\left\lfloor \log_2 m \right\rfloor} w^k D_k(m)$$ where $D_k(m)$ is the number of ways to decompose the integer $m$ into a product of $k$ factors greater than or equal to 2, the order of these factors being important.
We denote $\log_2$ the logarithm function in base 2, defined by $\log_2(x) = \frac{\ln(x)}{\ln(2)}$ for all real $x > 0$. With $w = \frac{1}{\lambda-1}$ and $\mathbf{f} = (1+w)\delta - w\mathbf{1}$, show that, for $s$ real sufficiently large,
$$\frac{1}{L_{\mathbf{f}}(s)} = 1 + \sum_{m=2}^{\infty} m^{-s} \sum_{k=1}^{\left\lfloor \log_2 m \right\rfloor} w^k D_k(m)$$
where $D_k(m)$ is the number of ways to decompose the integer $m$ into a product of $k$ factors greater than or equal to 2, the order of these factors being important.