One knows the Wikipedia article, this, and the MathWorld article, thisone dedicated to the Stirling numbers of the first kind. I've denoted $S^{k}(n)=s(n,k)$ as the signed Stirling numbers of first kind and thus $|s(n,k)|=\begin{bmatrix}n\\k\\\end{bmatrix}$ are the unsigned numbers, where $|x|$ is the absolute value of $x$.
In the recent past I did more experiments with the help of Wolfram Alpha online calculator, and I wondered if some of my experiments/claims can be proved as right or wrong. My experiments were involving these sequence of numbers and number theoretic functions.
I need an example to know how work with these series, because I don't know how to combine the asymptotic of Stirling numbers of first kind to deduce the convergence of my series.
Experiment 1. I believe that the following series involving signed Stirling numbers of first kind $$\sum_{n=1}^\infty\frac{S^{(\varphi(n))}_n}{n!}=\sum_{n=1}^\infty\frac{s(n,\varphi(n))}{n!}\tag{1}$$ and $$\sum_{n=1}^\infty\frac{S^{(\pi(n))}_n}{n!}=\sum_{n=1}^\infty\frac{s(n,\pi(n))}{n!}\tag{2}$$ are convergent series, where $\varphi(n)$ is the Euler's totient function and $\pi(x)$ denotes the prime-counting function. (If I remember well the same claims should be true with unsigned numbers $|s(n,k)|$ instead of $s(n,k)$ in previous series).$\square$
Experiment 2. Here I wanted to search a different combination, now involving in the summation the arithmetic function $\mu(n)$, the so-called Möbius function. It is an erratic function famous because is involved also in unsolved problems. I believe that $$\sum_{n=1}^\infty\frac{1}{n!}\sum_{m=1}^n S^{(m)}_n m^{1+\delta}\mu(m)\tag{3}$$ converges if $\delta=0$ and doesn't converge for each $\delta>0$.
This was
sum StirlingS1[n, m] m mu(m)/n!, from m=1 to n, from n=1 to 300
the code for this last experiment.$\square$
Question. Prove or refute the convergence of one of previous examples, to know how works with this kind of series. Many thanks.
Only is required an example with details/hints to deduce your result.
Of course with my scarced experiments and knowledges I don't know if my series are the more interesting for which one can ask*.
I wanted write this post as companion of a previous that I've asked (see [1]), being answered with a great answer.
References:
[1] Does $\sum_{n=1}^\infty \frac{1}{n!}{n\brace \varphi(n)}$ converge, where ${n\brace m}$ are Stirling numbers of second kind?, from this Mathematics Stack Exchange (2017).