# Question on conjectured formula for modular discriminant $\Delta(\tau)$ evaluated at $\tau=i\,\sigma$

This question assumes the definition of the modular discriminant $$\Delta(\tau)$$ in (1) below with Fourier series representation defined in (2) below where $$\tau(n)$$ is the Ramanujan tau function.

(1) $$\quad\Delta(\tau)=q\prod\limits_{r=1}^\infty(1-q^r)^{24}\,,\quad q=e^{2 \pi i \tau}$$

(2) $$\quad\Delta(\tau)=\sum\limits_{n=1}^\infty\tau(n)\,e^{2 \pi i n \tau}$$

I believe $$\Delta(i \sigma)$$ can be evaluated via formula (2) below where $$b(n)$$ is the Dirichlet convolution of $$\tau(n)$$ and $$\mu(n)$$, i.e. $$b(n)=\tau(n)*\mu(n)=\sum\limits_{d|n}\tau(d)\,\mu\left(\frac{n}{d}\right)$$.

(3) $$\quad\Delta(i \sigma)=\sum\limits_{n=1}^\infty b(n)\,(\coth(\pi n \sigma)-1)\,,\quad\sigma>0$$

The following figure illustrates formula (3) above for $$\Delta(i \sigma)$$ in orange overlaid on the reference function $$\Delta(i \sigma)=\eta(i \sigma)^{24}$$ in blue where the series in formula (3) above is evaluated over the first $$200$$ terms.

Figure (1): Illustration of formula (3) for $$\Delta(i \sigma)$$

More generally if $$f(\tau)$$ is a modular form with Fourier series illustrated in (4) below, I believe $$f(i \sigma)$$ can be evaluated via formula (5) below where $$b(n)$$ is the Dirichlet convolution of $$a(n)$$ and $$\mu(n)$$, i.e. $$b(n)=a(n)*\mu(n)=\sum\limits_{d|n}a(d)\,\mu\left(\frac{n}{d}\right)$$.

(4) $$\quad f(\tau)=a(0)+\sum\limits_{n=1}^\infty a(n)\,e^{2 \pi i n \tau}$$

(5) $$\quad f(i \sigma)=a(0)+\sum\limits_{n=1}^\infty b(n)\,(\coth(\pi n \sigma)-1)\,,\quad\sigma>0$$

Question: Is formula (3) for $$\Delta(i \sigma)$$ above (and the generalization to other modular forms in formula (5) above) a known and proven relationship and if so, can a reference be provided?

Sure expand $$\coth(\pi n \sigma)-1$$ into a geometric series in $$e^{2i\pi n \sigma}$$