Question related to nested Fourier series representation of $h(s)=\frac{i s}{s^2-1}$

In this question I use the term "nested Fourier series representation" to refer to an infinite series of Fourier series versus a single Fourier series. Whereas a single Fourier series is periodic, an infinite series of Fourier series is not necessarily periodic. When it is periodic it be expressed as a single Fourier series, and therefore doesn't provide as much increased utility.

This question is related to the function $$h(s)$$ defined in (1) below and its nested Fourier series representation defined in (2) below where the parameter $$f$$ is the evaluation frequency and assumed to be a positive integer. I believe formula (2) below converges for $$\Im(s)>0$$ but can be be analytically continued to the entire complex plane as illustrated in formula (3) below except at $$s=0$$ where $$\cot(0)$$ exhibits a complex infinity. Formulas (2) and (3) are both conditionally convergent and must be evaluated at $$M(N)=0$$ where $$M(x)$$ is the Mertens function. Since $$\sum_{n=1}^\infty\frac{\mu(n)}{n}=0$$, formula (3) can be simplified as illustrated in formula (4) below.

(1) $$\quad h(s)=\frac{i s}{s^2-1}$$

(2) $$\quad h(s)=\underset{N\to\infty\land f\to\infty}{\text{lim}}\left(\pi\sum_{n=1}^N\frac{\mu(n)}{n}\sum\limits_{k=1}^{f n} e^{\frac{2 \pi i k s}{n}}\right),\quad\Im(s)>0$$

(3) $$\quad h(s)=\underset{N\to\infty}{\text{lim}}\left(\frac{\pi i}{2}\sum\limits_{n=1}^N\frac{\mu(n)}{n}\left(\cot\left(\frac{\pi s}{n}\right)+i\right)\right),\quad s\ne 0$$

(4) $$\quad h(s)=\underset{N\to\infty}{\text{lim}}\left(\frac{\pi i}{2}\sum\limits_{n=1}^N\frac{\mu(n)}{n}\cot\left(\frac{\pi s}{n}\right)\right),\quad s\ne 0$$

The Laplace transform $$\mathcal{L}_t[h(i t)](z)$$ related to formulas (1) and (2) above are defined in formulas (5) and (6) below which must also be evaluated at $$M(N)=0$$ where $$\text{Si}(z)$$ and $$\text{Ci}(z)$$ are the sin and cos integral functions. I find it interesting a fairly complicated function such as (5) below can have a fairly simple formula such as (6) below. I also find the convergence of formula (6) below interesting as it seems to converge everywhere except at $$z=0$$ and along the negative real axis which I believe is related to a branch point at $$z=0$$ and a branch cut along the interval $$(-\infty,0)$$ of the $$\text{Ci}(z)$$ function.

(5) $$\quad\mathcal{L}_t[h(i t)](z)=\frac{\sin(z)}{2}(\pi-2\,\text{Si}(z))-\cos(z)\,\text{Ci}(z)$$

(6) $$\quad\mathcal{L}_t[h(i t)](z)=\underset{N\to\infty\land f\to\infty}{\text{lim}}\left(\pi\sum\limits_{n=1}^N\mu(n)\sum\limits_{k=1}^{f n}\frac{1}{2 \pi k+n z}\right),\quad\Re(z)>0\lor\Im(z)\neq 0$$

With respect to the Mellin transform $$\mathcal{M}_t[h(i t)](z)=\frac{\pi}{2}\sec \left(\frac{\pi z}{2}\right)$$ associated with formula (1) above, I'll note that the corresponding Mellin transforms related to formulas (2) and (3) above can both be shown to be equivalent to this Mellin tranform via analytic continuation and the definition of the Riemann zeta functional equation.

Formulas (2), (3), and (6) defined above are illustrated following the questions below.

In general I believe every function of the form illustrated in formula (7) below has a nested Fourier series representation of the form illustrated in formula (8) below where the parameter $$f$$ is the evaluation frequency and assumed to be a positive integer. Formula (8) below is typically valid for $$x>0$$ but in some cases the evaluation parameter $$N$$ can be selected to obtain convergence at $$x=0$$ in which case $$\overset{\text{~}}{f}(x)$$ evaluates to an odd function of $$x$$.

(7) $$\quad f(x)=\sum\limits_{n=1}^x a(n)\,,\quad a(n)\in\mathbb{C}$$

(8) $$\quad\overset{\text{~}}{f}(x)=\underset{N\to\infty\land f\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N b(n)\left(\frac{x}{n}-\left(\frac{1}{2}-\frac{1}{\pi}\sum\limits_{k=1}^{f\,n}\frac{\sin(\frac{2 k \pi x}{n})}{k}\right)\right)\right),\quad b(n)=\sum\limits_{d|n}a(d)\,\mu\left(\frac{n}{d}\right)$$

I believe the nested Fourier series representation of $$U(x)=\sum_{n=1}^x\delta_{n,1}=\theta(x-1)$$ is perhaps the most important example of (7) and (8) above because it's derivative $$U'(x)=\delta(x-1)$$ can be used to evaluate Mellin convolutions such as $$g(y)=\int_0^\infty\delta(x-1)\,g\left(\frac{y}{x}\right)\frac{dx}{x}$$ and $$g(y)=\int_0^\infty\delta(x-1)\,g(y\,x)\,dx$$ to derive new formulas for a variety of functions thereby providing new insights into functions and their relationships. Formula (1) above was actually derived in this manner but also involved a variable substitution. The reason for the conditional convergence requirement of formulas (2) to (4) and (6) above is because the nested Fourier series representation of $$\delta(x-1)$$ only converges at $$x=0$$ when $$M(N)=0$$. This answer I posted to my own question on the relationship between distributional and nested Fourier series frameworks for prime counting functions provides more information on this topic.

Formula (2) above illustrates there are nested Fourier series representations for non-periodic functions other than the form illustrated in formula (7) above. Note formulas (2) and (6) above are equivalent to formulas (9) and (10) below with $$g(s)=\frac{i s}{s^2-1}$$, $$c(n)=\pi\frac{\mu(n)}{n}$$, and $$d(k)=1$$. In the event $$d(k)=1$$ as was the case for $$h(s)$$ defined above, I believe formula (9) below can be analytically continued to the entire complex plane except $$s=0$$ as illustrated in formula (11) below. I suspect the conditional convergent requirement $$M(N)=0$$ stated above for formulas related to $$h(s)$$ may also apply to formulas (9) to (11) below for other examples of $$g(s)$$.

(9) $$\quad g(s)=\underset{N\to\infty\land f\to\infty}{\text{lim}}\left(\sum_{n=1}^N c(n)\sum\limits_{k=1}^{f n} d(k) e^{\frac{2 \pi i k s}{n}}\right),\quad\Im(s)>0$$

(10) $$\quad\mathcal{L}_t[g(i t)](z)=\underset{N\to\infty\land f\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N n\,c(n)\sum\limits_{k=1}^{f n} d(k)\frac{1}{2 \pi k+n z}\right)$$

(11) $$\quad g(s)=\underset{N\to\infty}{\text{lim}}\left(\frac{i}{2}\sum\limits_{n=1}^N c(n)\left(\cot\left(\frac{\pi s}{n}\right)+i\right)\right),\quad d(k)=1\land s\ne 0$$

Question: Are there other examples of non-periodic functions with nested Fourier series representations of the form illustrated in (9) above with Laplace transform $$\mathcal{L}_t[g(i t)](z)$$ illustrated in (10) above where $$g(s)$$ has a closed-form expression?

With respect to the question above, I'm not interested in functions trivially related to $$h(s)$$ defined in (1) above such as $$g(s)=A\,h(s)$$ where the constant $$A\in\mathbb{C}$$, or nested Fourier series representations of the integral and derivatives of $$h(s)$$ which can derived directly from formula (2) above.

In Figures (1) to (9) below, formulas (2), (3), and (6) defined above are illustrated in orange overlaid on the reference functions defined in formulas (1) and (5) above which are illustrated in blue. Formulas (2) and (6) seem to be much more sensitive to the magnitude of $$N$$ than the magnitude of $$f$$, and consequently all plots of formulas (2) and (6) illustrated below are evaluated at $$f=4$$ and $$N=214$$. All plots of formula(3) illustrated below are evaluated at $$N=401$$. These evaluation limits were selected as a trade-off between the conflicting goals of minimizing evaluation time and maximizing convergence. Note $$M(214)=M(401)=0$$ consistent with the conditional convergence requirement stated above.

The following figure illustrates formula (2) for $$f(s)$$ evaluated along the line $$s=i t$$ using the evaluation limits $$f=4$$ and $$N=214$$.

Figure (1): Illustration of formula (2) for $$f(s)$$ evaluated along the line $$s=i t$$

The following two figures illustrate the real and imaginary parts of formula (2) for $$f(s)$$ evaluated along the line $$s=t+i$$ using the evaluation limits $$f=4$$ and $$N=214$$.

Figure (2): Illustration of real part of formula (2) for $$f(s)$$ evaluated along the line $$s=t+i$$

Figure (3): Illustration of imaginary part of formula (2) for $$f(s)$$ evaluated along the line $$s=t+i$$

The following figure illustrates formula (3) for $$f(s)$$ evaluated along the line $$s=i t$$ using the evaluation limit $$N=401$$.

Figure (4): Illustration of formula (3) for $$f(s)$$ evaluated along the line $$s=i t$$

The following two figures illustrate the real and imaginary parts of formula (3) for $$f(s)$$ evaluated along the line $$s=t-i$$ using the evaluation limit $$N=401$$.

Figure (5): Illustration of real part of formula (3) for $$f(s)$$ evaluated along the line $$s=t-i$$

Figure (6): Illustration of imaginary part of formula (3) for $$f(s)$$ evaluated along the line $$s=t-i$$

The following figure illustrates formula (6) for $$\mathcal{L}_t[h(i t)](z)$$ evaluated for $$z\in\mathbb{R}$$ using the evaluation limits $$f=4$$ and $$N=214$$.

Figure (7): Illustration of formula (6) for $$\mathcal{L}_t[h(i t)](z)$$ evaluated for $$z\in\mathbb{R}$$

The following two figures illustrate the real and imaginary parts of formula (6) for $$\mathcal{L}_t[h(i t)](z)$$ evaluated along the line $$z=-1+i t$$ using the limits $$f=4$$ and $$N=214$$. Note formula (6) actually seems to converge better than the blue reference function defined in formula (5) as the magnitude of $$t$$ increases where both formulas are evaluated using Mathematica's default precision.

Figure (8): Illustration of real part of formula (6) for $$\mathcal{L}_t[h(i t)](z)$$ evaluated along the line $$z=-1+i t$$

Figure (9): Illustration of imaginary part of formula (6) for $$\mathcal{L}_t[h(i t)](z)$$ evaluated along the line $$z=-1+i t$$

The following figure illustrates a density plot of the absolute value of the difference between formulas (6) and (5) for $$\mathcal{L}_t[h(i t)](z)$$ where formula (6) is evaluated at $$f=4$$ and $$N=214$$. Note formula (6) seems to converge everywhere except at $$z=0$$ and along the interval $$(-\infty,0)$$ which I believe is related to a branch point and branch cut of the $$\text{Ci}(z)$$ function.

Figure (10): Density plot of the absolute value of the difference between formulas (6) and (5) for $$\mathcal{L}_t[h(i t)](z)$$

• Connection with "almost periodic functions" ? Feb 19, 2020 at 0:14
• Why don't you learn how to make your question short, simple, clear. In this form your question doesn't fit the needs Feb 19, 2020 at 3:23
• @reuns I provided background information believing it would help others to understand my question, but from your answer it appears you didn't. Feb 19, 2020 at 3:53

Iff $$\sum_{n\ge 1} a(n) n^{-2}$$ converges then $$h(x)=\sum_{n\ge 1} a(n)(\frac1{x+n}+\frac1{x-n})$$ converges and is meromorphic
Let $$f(x)= \sum_{n\ge 1} (\frac1{x+n}+\frac1{x-n})= \frac{2i\pi}{e^{2i\pi x}-1}-\frac1x+i\pi = \pi \cot (\pi x)-\frac{1}{x} = \frac{x}6 + O(x^3)$$ Then $$h(x)=\sum_{n\ge 1} b(n) f(x/n), \qquad a(n)=\sum_{d| n} d\, b(d), \qquad b(n) = \frac1n \sum_{d |n} \mu(d) a(n/d)$$ iff $$\sum_{n\ge 1} b(n) f(x/n)$$ converges iff $$\sum_{n\ge 1} b(n) n^{-1}$$ converges.
• It seems your answer can perhaps be used to provide a proof of formula (3) in my question above which is of some interest to me, but what I'm really looking for with this question is more examples of functions which both have a closed form expression and can also be expressed as a nested Fourier series as defined in formula (9) in my question above. The function $h(s)=\frac{i s}{s^2-1}$ defined in formula (1) of my question is the first example I found, but I'm looking for additional examples. Feb 19, 2020 at 23:17
• The function $h(s)=\frac{i s}{s^2-1}$ has a symmetry shared with modular forms of weight 0 which is $h\left(-\frac{1}{z}\right)=z^0 h(z)$. Do you think this symmetry is of any importance with respect to narrowing the search for additional examples? Feb 19, 2020 at 23:17