Jtdylktuy

This question assumes the following definitions.

(1) $quadpsi(x)=sumlimits_{n=1}^infty e^{-pi,n^2,x}=frac{1}{2} left(vartheta_3left(0,e^{-pi,x}right)-1right)$

(2) $quad f(x)=sumlimits_{n=1}^infty e^{-frac{pi,n^2}{x^2}}=frac{1}{2}left(vartheta_3left(0,e^{-frac{pi}{x^2}}right)-1right)$

(3) $quad M(K)=sumlimits_{n=1}^Kmu(k)qquadtext{(Mertens function)}$

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) $quadfrac{2,psi(x)+1}{2,psi(1/x)+1}=frac{1}{sqrt{x}}$

(5) $quadxi(s)=frac{1}{2}-frac{s,(1-s)}{2}sumlimits_{n=1}^inftyleft(left(sqrt{pi},nright)^{-s},Gammaleft(frac{s}{2},pi,n^2right)+left(sqrt{pi},nright)^{-(1-s)},Gammaleft(frac{1-s}{2},pi,n^2right)right)$

(6) $quadxi(s)=frac{1}{2}-frac{s,(1-s)}{2}sumlimits_{n=1}^inftyleft(E_{frac{1+s}{2}}left(pi,n^2right)+E_{frac{1+(1-s)}{2}}left(pi,n^2right)right)$

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) $quadxi(s)=s,(s-1)intlimits_0^infty f(x),x^{-s-1},dx=pi^{-frac{s}{2}}(s-1,)Gammaleft(frac{s}{2}+1right)sumlimits_{n=1}^inftyfrac{1}{n^s},,quadRe(s)>1$

(8) $quadfrac{2,f(x)+1}{2,fleft(frac{1}{x}right)+1}=x$

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) $quad e^{-frac{pi,n^2}{x^2}}=frac{x}{n}sum_{k=1}^Kfrac{mu(k)}{k},fleft(frac{n,k}{x}right),quad x>0land M(K)=0land Ktoinfty$

(10) $quad f(x)=xsumlimits_{n=1}^inftyfrac{1}{n}sumlimits_{k=1}^Kfrac{mu(k)}{k},fleft(frac{n,k}{x}right),quad x>0land M(K)=0land Ktoinfty$

Relationship (10) above can be written in terms of the Jacobi theta function as follows.

(11) $quadvartheta _3left(0,e^{-pi,x^2}right)-1=frac{1}{x}sumlimits_{n=1}^inftyfrac{1}{n}sumlimits_{k=1}^inftyfrac{mu(k)}{k}left(vartheta_3left(0,e^{-frac{pi}{n^2,k^2,x^2}}right)-1right),quad x>0$

(12) $quadvartheta_3left(0,e^{-frac{pi}{x^2}}right)-1=xsumlimits_{n=1}^inftyfrac{1}{n}sumlimits_{k=1}^{infty}frac{mu(k)}{k}left(vartheta_3left(0,e^{-frac{pi,x^2}{n^2,k^2}}right)-1right),quad x>0$

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) $quadvartheta _3left(0,e^{-pi,x^2}right)-1=frac{1}{x}sumlimits_{n=1}^Nfrac{1}{n}sumlimits_{k=1}^Kfrac{mu(k)}{k}left(vartheta_3left(0,e^{-frac{pi}{n^2,k^2,x^2}}right)-1right),\$ $qquadqquadqquadqquadqquadqquadqquadqquadqquadqquad x>0land Ntoinftyland M(K)=0land Ktoinfty$

(14) $quadvartheta_3left(0,e^{-frac{pi}{x^2}}right)-1=xsumlimits_{n=1}^Nfrac{1}{n}sumlimits_{k=1}^Kfrac{mu(k)}{k}left(vartheta_3left(0,e^{-frac{pi,x^2}{n^2,k^2}}right)-1right),\$ $qquadqquadqquadqquadqquadqquadqquadqquadqquadqquad x>0land Ntoinftyland M(K)=0land Ktoinfty$

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) $qquadvartheta_3(z,tau)=(-i,tau)^{-frac{1}{2}},e^{frac{z^2}{pi,i,tau}},vartheta_3left(frac{z}{tau},-frac{1}{tau}right)$

Question (1): Are there any other interesting and unique relationships that have been or can be derived related to the Jacobi theta functions $vartheta _3left(0,e^{-pi,x^2}right)$ and $vartheta_3left(0,e^{-frac{pi}{x^2}}right)$? Do these functions play any special role in applications of the Jacobi theta function or related theory such as the theory of modular forms?