This question assumes the following definitions.
(1) $\quad\psi(x)=\sum\limits_{n=1}^\infty e^{-\pi\,n^2\,x}=\frac{1}{2} \left(\vartheta_3\left(0,e^{-\pi\,x}\right)-1\right)$
(2) $\quad f(x)=\sum\limits_{n=1}^\infty e^{-\frac{\pi\,n^2}{x^2}}=\frac{1}{2}\left(\vartheta_3\left(0,e^{-\frac{\pi}{x^2}}\right)-1\right)$
(3) $\quad M(K)=\sum\limits_{n=1}^K\mu(k)\qquad\text{(Mertens function)}$
Riemann used the Jacobi theta functional equation in the form illustrated in (4) below to prove the Riemann Xi functional equation $\xi(s)=\xi(1-s)$ (e.g. see section 1.7 of "Riemann's Zeta Function" by H. M. Edwards). The functional equation illustrated in (4) below was used to derive the two equivalent formulas for $\xi(s)$ illustrated in (5) and (6) below which I believe are globally convergent. Note both of these formulas are unchanged by the substitution $s=1-s$.
(4) $\quad\frac{2\,\psi(x)+1}{2\,\psi(1/x)+1}=\frac{1}{\sqrt{x}}$
(5) $\quad\xi(s)=\frac{1}{2}-\frac{s\,(1-s)}{2}\sum\limits_{n=1}^\infty\left(\left(\sqrt{\pi}\,n\right)^{-s}\,\Gamma\left(\frac{s}{2},\pi\,n^2\right)+\left(\sqrt{\pi}\,n\right)^{-(1-s)}\,\Gamma\left(\frac{1-s}{2},\pi\,n^2\right)\right)$
(6) $\quad\xi(s)=\frac{1}{2}-\frac{s\,(1-s)}{2}\sum\limits_{n=1}^\infty\left(E_{\frac{1+s}{2}}\left(\pi\,n^2\right)+E_{\frac{1+(1-s)}{2}}\left(\pi\,n^2\right)\right)$
The relationship between $f(x)$ illustrated in (2) above and the Riemann Xi function $\xi(s)$ is illustrated in (7) below. The Jacobi theta functional equation related to $f(x)$ is illustrated in (8) below, but this functional equation was not used in the derivation of formula (7) below.
(7) $\quad\xi(s)=s\,(s-1)\int\limits_0^\infty f(x)\,x^{-s-1}\,dx=\pi^{-\frac{s}{2}}(s-1\,)\Gamma\left(\frac{s}{2}+1\right)\sum\limits_{n=1}^\infty\frac{1}{n^s}\,,\quad\Re(s)>1$
(8) $\quad\frac{2\,f(x)+1}{2\,f\left(\frac{1}{x}\right)+1}=x$
Since the formulas for $\xi(s)$ derived from $\psi(x)$ are valid for all s, whereas the formula for $\xi(s)$ derived from $f(x)$ is only valid for $\Re(s)>1$, it would seem that $\psi(x)$ is perhaps a more important function than $f(x)$. Nevertheless, I've found $f(x)$ to be a very interesting function primarily because it obeys the relationships illustrated in (9) and (10) below. Below I indicated the two relationships are valid for $x>0$, but I actually think they're valid for a subset of $Re(x)>0$ (possibly $|\Im(x)|<|\Re(x)|$).
(9) $\quad e^{-\frac{\pi\,n^2}{x^2}}=\frac{x}{n}\sum_{k=1}^K\frac{\mu(k)}{k}\,f\left(\frac{n\,k}{x}\right),\quad x>0\land M(K)=0\land K\to\infty$
(10) $\quad f(x)=x\sum\limits_{n=1}^\infty\frac{1}{n}\sum\limits_{k=1}^K\frac{\mu(k)}{k}\,f\left(\frac{n\,k}{x}\right),\quad x>0\land M(K)=0\land K\to\infty$
Relationship (10) above can be written in terms of the Jacobi theta function as follows.
(11) $\quad\vartheta _3\left(0,e^{-\pi\,x^2}\right)-1=\frac{1}{x}\sum\limits_{n=1}^\infty\frac{1}{n}\sum\limits_{k=1}^\infty\frac{\mu(k)}{k}\left(\vartheta_3\left(0,e^{-\frac{\pi}{n^2\,k^2\,x^2}}\right)-1\right),\quad x>0$
(12) $\quad\vartheta_3\left(0,e^{-\frac{\pi}{x^2}}\right)-1=x\sum\limits_{n=1}^\infty\frac{1}{n}\sum\limits_{k=1}^{\infty}\frac{\mu(k)}{k}\left(\vartheta_3\left(0,e^{-\frac{\pi\,x^2}{n^2\,k^2}}\right)-1\right),\quad x>0$
When evaluated at finite limits formulas (11) and (12) above are conditionally convergent with respect to the inner sum over $k$ and must be evaluated as follows.
(13) $\quad\vartheta _3\left(0,e^{-\pi\,x^2}\right)-1=\frac{1}{x}\sum\limits_{n=1}^N\frac{1}{n}\sum\limits_{k=1}^K\frac{\mu(k)}{k}\left(\vartheta_3\left(0,e^{-\frac{\pi}{n^2\,k^2\,x^2}}\right)-1\right),\\$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad x>0\land N\to\infty\land M(K)=0\land K\to\infty$
(14) $\quad\vartheta_3\left(0,e^{-\frac{\pi}{x^2}}\right)-1=x\sum\limits_{n=1}^N\frac{1}{n}\sum\limits_{k=1}^K\frac{\mu(k)}{k}\left(\vartheta_3\left(0,e^{-\frac{\pi\,x^2}{n^2\,k^2}}\right)-1\right),\\$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad x>0\land N\to\infty\land M(K)=0\land K\to\infty$
I've read the functional equation illustrated in (15) below is of considerable importance in mathematics with far-reaching consequences, and I'm wondering if the relationships illustrated in (13) and (14) above are also perhaps of some significance.
(15) $\qquad\vartheta_3(z,\tau)=(-i\,\tau)^{-\frac{1}{2}}\,e^{\frac{z^2}{\pi\,i\,\tau}}\,\vartheta_3\left(\frac{z}{\tau},-\frac{1}{\tau}\right)$
Question (1): Are there any other interesting and unique relationships that have been or can be derived related to the Jacobi theta functions $\vartheta _3\left(0,e^{-\pi\,x^2}\right)$ and $\vartheta_3\left(0,e^{-\frac{\pi}{x^2}}\right)$? Do these functions play any special role in applications of the Jacobi theta function or related theory such as the theory of modular forms?