TITLE: Have any proposals been advanced for the analytic continuation of the divisor function?
QUESTION [11 upvotes]: While I was working on the evaluation of a certain series, the following limit came up:
\begin{align} \lim_{n \to 1} \frac{d(n)-1}{n(n-1)} &= \lim_{n \to 1} \frac{d'(n)}{2n-1} \\
&= d'(1)  .\end{align}
Here, I used l'Hôpitals rule, and $d(\cdot)$ denotes the divisor function. In order to compute the derivative of a function, one must know how it is defined on the real numbers.  Unfortunately, I have not found any proposals that described the notion of the analytic continuation for the divisor function so far. It seems that it is defined on $\mathbb{Z}$ only.
I did find identities for $\sigma_{\alpha} (x)$ for any $\alpha \in \mathbb{C}$ and $x \in \mathbb{Z}_{\geq 1}$. However, this is not what I am looking for. I seek to find a suitable expression for $\sigma_{\alpha} (x)$ when $\alpha = 0$, and real or complex arguments $x$.
I believe it may be possible to find such an extension of the divisor function, in part because such extensions have been found for the lowercase prime omega function. This arithmetic function is related to the divisor function. The continuation of the function is as follows: $$ \DeclareMathOperator{\sinc}{sinc} \omega(z) = \log_{2} \Bigg{(} \sum_{x=1}^{\lceil Re(z) \rceil} \sinc \Bigg{(} \prod_{y=1}^{\lceil Re(z) \rceil + 1 } (x^{2} + x - yz) \Bigg{)} \Bigg{)} ,$$ where $\sinc(\cdot)$ is the normalized sinc function.
Question: have any proposals been advanced for the analytic continuation of the divisor function, thereby extending the domain to $\mathbb{R}$ or $\mathbb{C}$ ?

REPLY [2 votes]: Since the formula I posted in my previous answer only converges as $N\to\infty$ at $x\in\mathbb{Z}_{\ne 0}$, I decided to post another formula which I believe converges as $N\to\infty$ for $x\in\mathbb{C}$. The previous answer I posted is consistent with the fact that zero has an infinite number of divisors, but this answer assumes the definition $\sigma_0(0)=0$.

Consider the following definitions of the analytic extension $\tilde{\sigma}_0(x)$ of $\sigma_0(n)$ and it's first order derivative $\tilde{\sigma}_0'(x)$ which are based on partial evaluations of real analytic formulas for $\tilde{f_{\sigma_0}}'(x)=\sum\limits_n \sigma_0(n)\,\delta(x-n)$ and $\tilde{f_{\sigma_0}}''(x)=\sum\limits_n \sigma_0(n)\,\delta'(x-n)$ (see this answer I posted to a related question on Math StackExchange).

$$\tilde{\sigma}_0(x)=\underset{N\to\infty}{\text{lim}}\left(\frac{1}{2}\sum\limits_{n=1}^N\left(-\frac{\sin(2 \pi x)}{\pi x}+\frac{1}{n}\sum\limits_{k=1}^n\left(\cos\left(\frac{2 \pi k x}{n}\right)+\cos\left(\frac{2 \pi (k-1) x}{n}\right)\right)\right)\right)\tag{1}$$
$$\tilde{\sigma}_0'(x)=\underset{N\to\infty}{\text{lim}}\left(\frac{1}{2}\sum\limits_{n=1}^N\left(\frac{\sin(2 \pi x)-2 \pi x \cos(2 \pi x)}{\pi x^2}-\frac{2 \pi}{n^2}\sum\limits_{k=1}^n\left(k \sin\left(\frac{2 \pi k x}{n}\right)+(k-1) \sin\left(\frac{2 \pi (k-1) x}{n}\right)\right)\right)\right)\tag{2}$$

Figure (1) below illustrates formula (1) for $\tilde{\sigma}_0(x)$ evaluated at $N=100$ in blue where the red discrete evaluation points illustrate $\sigma_0(x)$ for $x=n\in\mathbb{Z}$.


Figure (1): Illustration of formula (1) for $\tilde{\sigma}_0(x)$

The following table illustrates formula (2) for $\tilde{\sigma}_0'(x)$ evaluated at $x=1$ seems to converge to approximately $-3.35$ as $N\to\infty$.

$$\begin{array}{cc}
 N & \tilde{\sigma}_0'(1) \\
 100 & -3.31334 \\
 1000 & -3.34278 \\
 10000 & -3.34574 \\
\end{array}$$