I am using the example from Theorem 1 of Friedrich Pillichshammer "Euler's Constant and Averages of Fractional Parts" (https://www.dmg.tuwien.ac.at/nfn/gamma.pdf) we have for integers $k > 1$ applying the dirichiet convolution theorem ($f \left({n}\right) = 1$)
(1) $$\sum_{n \le {x}^{1/k}} Fraction \left(\frac{x}{{n}^{k}}\right) = x \sum_{n \le {x}^{1/k}} \frac{1}{n^k} - \sum_{n \le x} f * 1 \left({n}\right) \sim - \left({\frac{1}{k - 1} + \zeta \left({\frac{1}{k}}\right)}\right) {x}^{1/k} + O \left({{x}^{1/\left({k + 1}\right)}}\right)$$
Where
$$\sum_{n \le x} f * 1 \left({n}\right) = \sum_{n \le y} \left\lfloor{\sqrt[k]{\frac{x}{m^k}}}\right\rfloor + \sum_{m \le \sqrt[k]{x/y}} \left\lfloor{\frac{x}{m^k}}\right\rfloor - \left\lfloor{\sqrt[k]{\frac{x}{y}}}\right\rfloor * \left\lfloor{y}\right\rfloor \sim \zeta \left({\frac{1}{k}}\right) \sqrt[k]{x} + \zeta \left({k}\right) x + O \left({\min \left({y, \sqrt[k]{\frac{x}{y}}}\right)}\right)$$
In the paper you can assign $y = \sqrt{x}$ for Eq (1) above or the final solution of $y = x^{1/(k+1)}$ for the solution that minimizes the error terms.
I am trying to apply the same hyperbola method to
$$\sum_{n \le {x}^{1/k}} n * Fraction \left(\frac{x}{{n}^{k}}\right)$$
To complete the asymptotic solution to
$$\sum_{n \le {x}^{1/k}} n * \left\lfloor\frac{x}{{n}^{k}}\right\rfloor$$
In trying to use the hyperbola method I get using $f \left({n}\right) = n$
$$\sum_{n \le x} f * 1 \left({n}\right) = \sum_{n \le y} \sum_{{m}^{k} \le x/n} n + \sum_{{m}^{k} \le x/y} m \left\lfloor{\frac{x}{{m}^{k}}}\right\rfloor - \left({\sum_{n \le y} n}\right) \left({\sum_{{m}^{k} \le x/y} 1}\right)$$
Which is not correct. It is not clear as to why I have this error. Note that for $k = 2$ we will have a $(1/2) x \log \left({x}\right)$ leading order term.
Simple error analysis results in
$$\sum_{n \le \left\lfloor{\sqrt[k]{x}}\right\rfloor} n\, \left\lfloor{\frac{x}{{n}^{k}}}\right\rfloor \sim \sum_{n \le \left\lfloor{\sqrt[k]{x}}\right\rfloor} n \left({\frac{x}{{n}^{k}} + O \left({1}\right)}\right) \sim \begin{cases} \frac{1}{2}\, x\, \log \left({x}\right) + \gamma\, x + O \left({x}\right), & k = 2, \\ \zeta \left({k - 1}\right) x - \frac{{x}^{2/k}}{k - 2} + O \left({{x}^{2/k}}\right), & k \ge 3. \end{cases}$$
However the error terms is equal to the next order term in the series. The goal is to refine this error using the hyperbola method or other methods.