First, let me introduce my little story:
I started searching information about the famous Dirichlet Divisor Problem: getting the exact asymptotic behaviour of the sum of the divisor function up to an integer $x$. More specifically, the aim of the problem is to bound $\theta$ in:
$$D(x)=\sum_{n \le x} d(n) = x \log x +x(2 \gamma -1) + O(x^{\theta + \epsilon})$$
Then, I found that this problem could be generalised to finding the sum of the number of ways an integer $n$ can be written as the product of $k$ natural numbers up to an integer $x$, which is called the Piltz Divisor Problem. Again, it is based on bounding the error term in:
$$D_k(x)=xP_k(\log x) + O(x^{\alpha_k + \epsilon})$$
being $P_k$ a polynomial of degree $k$.
After all that, I found a paper relating these two problems to the Riemann Hypothesis, which seemed something quite interesting for me. It was called The Generalized Divisor Problem and The Riemann Hypothesis, by Hideki Nakaya. It starts quoting in (1) a work by A. Selberg (which I have not been able to find) where he extended Piltz's Divisor Problem to all complex $k$. And I have a lot of trouble when trying to understand this.
I. How can one extend Piltz's Problem to complex numbers?
II. What would that 'extension' be useful for?
III. Does this 'extension' have any geometrical interpretations such as the hyperbola method of both Dirichlet's and Piltz's Problems?
IV. Is there any direct relationship between the error bound on Piltz's Problem and the error bound for the extended problem?
I know these are a lot of questions, so thank you for your help.