I am interested in the function ${f_n}\left( m \right)$ which can be defined by the Dirichlet generating function $$\zeta \left( s \right)\zeta \left( {s - 1} \right)...\zeta \left( {s - n + 1} \right) = \sum\limits_{m = 1}^\infty {\frac{{{f_n}\left( m \right)}}{{{m^s}}}} $$

This is called the number-of-sublattices function. It is straightforward to get the standard estimate of $\sum\limits_{m \leqslant x}^{} {{f_n}\left( m \right)}$:

$$\sum\limits_{m \leqslant x}^{} {{f_n}\left( m \right)} = \frac{{{x^n}}}{n}\prod\limits_{\ell = 2}^{n - 1} {\zeta \left( \ell \right)} + O\left( {{x^{n - 1}}\log x} \right)\quad \quad \quad \left( 1 \right)$$

I am looking for an improvement in the error term. The sharpest known estimate for $\sum\limits_{m \leqslant x}^{} {{\sigma _1}\left( m \right)}$ is known due to Walfisz:

$$\sum\limits_{m \leqslant x}^{} {{\sigma _1}\left( m \right)} = \frac{{{\pi ^2}}}{{12}}{x^2} + O\left( {x{{\log }^{2/3}}x} \right)\quad \quad \quad \left( 2 \right)$$

Since $\zeta \left( s \right)\zeta \left( {s - 1} \right) = \sum\limits_{m = 1}^\infty {\frac{{{\sigma _1}\left( m \right)}}{{{m^s}}}}$, I have been wondering if it is possible to get a sharper bound for the error term of (1) using (2).

So far, I have tried estimating \${f_3}\left( m \right) = \sum\limits_{d|m}^{} {{\sigma _1}\left( d \right){{\left( {\tfrac{m}{d}} \right)}^2}} \$ but without success.

I have tried summation interchange identities as well as the Dirichlet hyperbola method. I also tried to do some work involving the Fourier series representation of the fractional part, again unsuccessfully. Does anybody know of any theorems or techniques which could be use (2) to obtain a better error term in (1)?