Extending Piltz's Divisor Problem to complex numbers

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.

Answer

Question I: Let $\mathbb{1}(n)=1$ for all $n$. As explained here, the divisor function can be expressed as $d(n)=\sum_{d|n}1=\mathbb{1}*\mathbb{1}$ (where $*$ denotes Dirichlet convolution), meaning that $$\sum_{n=1}^\infty\frac{d(n)}{n^s}=\Big{(}\sum_{n=1}^\infty\frac{1}{n^s}\Big{)}\Big{(}\sum_{n=1}^\infty\frac{1}{n^s}\Big{)}=\zeta^2(s)$$ In the paper which you cite, the divisor function is defined using the expression above instead of the more common "sum of divisors" idea. Indeed, the paper opens by stating that $d_z(n)$ is defined as the function which satisfies $$\zeta^z(s)=\sum_{n=1}^\infty\frac{d_z(n)}{n^s}$$So, when $z=2$, we obtain the divisor function descibed above, but this is not true for other values of $z$. Indeed, for positive integers $z$, we find that $d_z(n)=\mathbb{1}*\mathbb{1}*...*\mathbb{1}$ ($z$ times; the reasoning behind this is explained here), which is not the same as the standard extensions of the divisor function. So, by generalizing to the complex numbers, we are essentially asking for the function $d_z(n)$ which will satisfy the equation above for some complex $z$.

Question II: This extension is useful as it gives us the tools to asymptotically estimate $\pi_k(n)$, which is defined as the number of integers less then $n$ which have at most $k$ prime factors. Note that $\pi_1(n)$ is simply the prime counting function. Landau showed in 1900 that $$\pi_k(n)\sim\frac{x (\ln\ln x)^{k-1}}{(k-1)!\ln x}$$When $k=1$, this reduces to the prime number theorem. Anyway, what the paper you cite explains is that Selberg thought to use the ideas of extending Plitz's Divisor Problem to obtain a more "accurate" asymptotic expansion. More about this is explained in the paper to which you refer.

It seems like questions $3$ and $4$ are partially addressed in the remainder of the paper, but it seems to get messier from then. I do not intend to dive in too deep into the paper, so I doubt I will be able to answer the remaining two questions, but I hope this begins to explain at least the first two.