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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.


Discrete plot of absolute value of formula (2) for f(s) evaluated along the real axis

Figure (1): Discrete plot of absolute value of formula (2) for $f(s)$ evaluated along the real axis


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.


Discrete plot of absolute value of formula (2) for f(s) evaluated along the line s=t-i\ t

Figure (3): Discrete plot of absolute value of formula (2) for $f(s)$ evaluated along the line $s=t-i\ t$

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  • $\begingroup$ Define 'the Dirac delta distribution (not function)' $\endgroup$
    – reuns
    May 6, 2020 at 1:25
  • $\begingroup$ @reuns I think your comment is not relevant to the question. $\endgroup$ May 6, 2020 at 1:40
  • $\begingroup$ @reuns This question is asking pointwise convergence of $\delta_N$ except some points. $\endgroup$ May 6, 2020 at 2:04
  • $\begingroup$ @JingeonAn That's the point of my comment : the OP doesn't see that it is not about Dirac delta at all, and I'm not sure you do. Since the OP works with Mellin transforms understanding convergence to $\delta$ and other distributions is very important for him. $\endgroup$
    – reuns
    May 6, 2020 at 2:11
  • $\begingroup$ @reuns I don't really see the point of your comment on the distributuion. The question is given explicitely, it is yes or no question (even though 'dirac delta' in the title may not be correct wording). Can you explain your point more explicitly? $\endgroup$ May 6, 2020 at 2:15

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