Formula (2) below for $f(s)$ was derived from formula (1) below for $s$ which is proven in this answer to this question on Math Overflow. Formula (2) was derived from formula (1) by evaluating the Mellin transform of the derivative of formula (1) followed by a variable substitution. I say $f(s)$ is related to the Dirac delta function $\delta(s)$ because this derivation is based on the relationship $\delta (s)=\frac{1}{2 \pi}\mathcal{M}_x[1](-i s)$. The $f(s)$ function may not be a valid representation for $\delta(s)$, but I believe it is similar to $\delta(s)$ in that $f(s)=\delta(s)=0$ for $s\in\mathbb{R}\land s\ne 0$.
(1) $\quad s=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N\frac{1}{n}\sum\limits_{k=0}^n (-1)^k \left( \begin{array}{c} n \\ k \\ \end{array} \right)\frac{1}{(k+1)^s}\right),\quad s\in\mathbb{C}$
(2) $\quad f(s)=\underset{N\to\infty}{\text{lim}}\left(\frac{\Gamma(-i\ s)}{2\ \pi}\sum\limits_{n=1}^N \frac{1}{n}\sum\limits_{k=1}^n (-1)^{k+1} \left( \begin{array}{c} n \\ k \\ \end{array} \right) (\log(k+1))^{1+i\ s}\right)$
Question: Is it true formula (2) above for $f(s)$ converges to zero globally except at $s=i\ n$ for some non-positive integer $n$ which corresponds to the poles of $\Gamma(-i\ s)$, or does formula (2) above converge to zero only for $s\in \mathbb{R}\land s\ne 0$.
It's very difficult to verify the convergence of the two formulas above observationally because they both converge very slowly at non-integer real values of $s$ much less complex values of $s$. This is also a characteristic of formula (3) below for $\zeta(s)$ which is the basis of formula (1) above for $s$ (see this question on Math StackExchange).
(3) $\quad\zeta(s)=\underset{N\to\infty}{\text{lim}}\left(\frac{1}{s-1}\sum\limits_{n=0}^N \frac{1}{n+1}\sum\limits_{k=0}^n (-1)^k \left( \begin{array}{c} n \\ k \\ \end{array} \right)\frac{1}{(k+1)^{s-1}}\right),\quad s\in\mathbb{C}$
The following two figures illustrate formula (2) for $f(s)$ converges much faster at integer values of $s$ than non-integer values of $s$. The plots in both figures were generated using $32$ digit precision with an evaluation limit of $N=100$ for formula (2) above.
Figure (1): Discrete plot of absolute value of formula (2) for $f(s)$ evaluated along the real axis
Figure (2): Plot of absolute value of formula (2) for $f(s)$ evaluated along the real axis
The following figure illustrates formula (2) for $f(s)$ may have a wider convergence than $s\in\mathbb{R}\land s\ne 0$. The discrete plot in the figure below was evaluated along the line $s=t-i\ t$ using $32$ digit precision with an upper evaluation limit of $N=100$ for formula (2) above.
Figure (3): Discrete plot of absolute value of formula (2) for $f(s)$ evaluated along the line $s=t-i\ t$