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Normal convergence allows termwise differentiation of the defining theta function with characteristics. A tau derivative multiplies a summand by $\pi i(n+\alpha {)}^2$, while its second spatial derivative multiplies it by $-4{\pi }^2(n+\alpha {)}^2$. This proves the displayed heat equation, linking its lattice-sum construction with diffusion and analytic continuation.

Use period-one spatial normalization and $Q=e^{\pi i\tau }$. The conventional odd function is ${\theta }_1=-\theta [1/2,1/2]$ with the characteristic convention of theta function with characteristics. Differentiating its Jacobi triple product gives ${\theta }_1^{\prime }(0)=2\pi Q^{1/4}{\prod }_{m\ge 1}(1-Q^{2m}{)}^3$. The three even constant products have product $2Q^{1/4}{\prod }_{m\ge 1}(1-Q^{2m}{)}^3$, proving the identity. The sign changes if one instead calls the characteristic series itself theta1.

For the binary characteristic convention, each theta function with characteristics has one simple zero per lattice cell. The four zeros of the numerator, at the four half-periods, cancel those of ${\vartheta }_{11}(2z)$. Their quotient is an entire elliptic function, hence constant. Taking the limit at zero determines the factor one half. The product identity remains valid at the zeros, where the initial quotient requires removable extension.

Fix $\tau \in \mathbb{H}$ and let $\Lambda =\mathbb{Z}+\mathbb{Z}\tau$. We use the binary convention for the theta function with characteristics in the PDF. Its nonzero exponential prefactor does not change the zeros of the translated ${\vartheta }_{00}$. The four characteristic functions are entire in $z$, by normal convergence of their Gaussian series.

We first prove their zero information, rather than assume that the quotient has no poles. Pair the terms with indices $n$ and $-n-1$ to getThe logarithmic derivative $D(z)={\vartheta }_{00}^{\prime }(z)/{\vartheta }_{00}(z)$ is periodic under $z\mapsto z+1$ and satisfies $D(z+\tau )=D(z)-2\pi i$, by the spatial transformation formulas. Choose a fundamental parallelogram with zero-free boundary. Its vertical-side integrals cancel, and subtracting the top horizontal integral from the bottom one gives $2\pi i$. The argument principle therefore counts exactly one zero inside, with multiplicity. Thus the displayed zero is simple and is the unique one modulo $\Lambda$. The zeros of theta functions with characteristics are consequentlyAll are simple. In particular the three even theta constants are nonzero and ${\vartheta }_{11}^{\prime }(0)\ne 0$.

Let $R(z)$ be the product of these four functions divided by ${\vartheta }_{11}(2z)$. The denominator's simple zeros occur exactly at the four half-periods modulo $\Lambda$, and each is cancelled by the corresponding simple numerator zero. Hence $R$ extends to an entire function. Under $z\mapsto z+1$, the four numerator signs multiply to one, as do the two successive shifts of the denominator. Under $z\mapsto z+\tau$, the numerator has multiplier $q^{-2}{t}^{-4}$, where $t=e^{2\pi iz}$; the denominator, whose argument shifts by $2\tau$, has the same multiplier. Thus $R$ is an entire elliptic function with periods $1,\tau$. It is bounded on a fundamental parallelogram and hence on the plane, so Liouville's theorem makes it constant.

At $z=0$, the simple odd zero gives ${\vartheta }_{11}(z)/{\vartheta }_{11}(2z)\to 1/2$. Therefore the Jacobi theta duplication identity isAt a zero of the displayed denominator, this is interpreted by removable continuation. Equivalently, multiply through by the denominator to obtain an entire product identity valid for every $z$.

State the characteristic convention explicitly: for real $\alpha ,\beta$, define the theta function with characteristics byBinary subscripts ${\vartheta }_{ab}$ often denote this function with $(\alpha ,\beta )=(a/2,b/2)$, $a,b\in \{0,1\}$. On compact subsets of the $(z,\tau )$ domain, a summand and any derivative are bounded by a polynomial in $|n|$ times $e^{-c{n}^2+C|n|}$. Normal convergence therefore gives an entire function of $z$, holomorphic dependence on $\tau$, and legitimate termwise differentiation. In particular the heat equation for theta functions with characteristics isCharacteristics are defined modulo integers with a phase: $\theta [\alpha +1,\beta ]=\theta [\alpha ,\beta ]$ and $\theta [\alpha ,\beta +1]=e^{2\pi i\alpha }\theta [\alpha ,\beta ]$. Shifting the summation index proves the spatial quasi-periodsThese multipliers describe a degree-one line bundle on the complex torus with periods $1,\tau$, rather than an ordinary elliptic function. The logarithmic derivative is unchanged under one and decreases by $2\pi i$ under tau. Pairing opposite sides of a fundamental cell therefore makes its contour integral $2\pi i$. The argument principle counts exactly one zero in a cell, with multiplicity.

The characteristic function is a translate of the unshifted theta function:At $z=(1+\tau )/2$, the unshifted series cancels in pairs $n,-n-1$. Hence the zeros of theta functions with characteristics occur exactly and simply atReindexing also gives $\theta [\alpha ,\beta ](-z)=\theta [-\alpha ,-\beta ](z)$. For half-integer characteristics the parity is $e^{4\pi i\alpha \beta }$, so only $(1/2,1/2)$ is odd and vanishes at zero; the other three theta constants are nonzero.

The modular transformations of theta characteristics are obtained from Gaussian Poisson summation and elementary phase rearrangement. With the holomorphic branch of $\sqrt{-i\tau }$ positive at $\tau =i$,For the first formula, apply Poisson summation to $F(t)=e^{\pi i\tau (t+\alpha {)}^2+2\pi i(t+\alpha )(z+\beta )}$. Completing the square in its Fourier transform givesRearranging ${\sum }_{n}F(n)={\sum }_m\hat{F}(m)$ proves the stated inversion law. For translation by one, use $e^{\pi i{n}^2}=e^{\pi in}$ in the defining series; the remaining characteristic phase is precisely the displayed factor.

Write $Q=e^{\pi i\tau }$, so the modular-form variable of Question 1 is $q=Q^2$, and put ${\Theta }_2=\theta [1/2,0](0,\tau )$, ${\Theta }_3=\theta [0,0](0,\tau )$ and ${\Theta }_4=\theta [0,1/2](0,\tau )$. Inversion interchanges ${\Theta }_2,{\Theta }_4$ and fixes ${\Theta }_3$, each with factor $\sqrt{-i\tau }$. Translation interchanges ${\Theta }_3,{\Theta }_4$ and multiplies ${\Theta }_2$ by $e^{\pi i/4}$. The Jacobi triple product givesIt also gives the conventional odd theta function ${\theta }_1=-\theta [1/2,1/2]$ its product $2Q^{1/4}\sin (\pi z){\prod }_{m\ge 1}(1-Q^{2m})(1-2Q^{2m}\cos (2\pi z)+Q^{4m})$. Differentiate at zero and multiply the three even products to obtain the Jacobi derivative formulaIn the constant-product simplification, ${\prod }_m(1+Q^{2m})(1-Q^{4m-2})=1$, by separating even and odd positive exponents. The products prove nonvanishing of all three even constants as well.

The Jacobi abstruse identity can be proved directly, without assuming it from the products. Splitting pairs of summation indices according to whether their sum and difference are even or odd givesFor example, $u=(n+m)/2,v=(n-m)/2$ are either both integers or both half-integers, and $n^2+m^2=2u^2+2v^2$. The alternating signs give the second identity; shifting both original indices by one half gives the third. Squaring the first two and subtracting then proves ${\Theta }_3^4={\Theta }_2^4+{\Theta }_4^4$.

These identities connect theta theory to modular forms and elliptic curves. The fourth powers are weight-two forms on the principal level-two congruence subgroup. The modular lambda function $\lambda ={\Theta }_2^4/{\Theta }_3^4$ obeys $\lambda (-1/\tau )=1-\lambda (\tau )$ and $\lambda (\tau +1)=\lambda (\tau )/(\lambda (\tau )-1)$; the characteristic permutation kernel is $\Gamma (2)$. It parametrizes an elliptic curve with its ordered two-torsion in Legendre form $y^2=x(x-1)(x-\lambda )$. The corresponding Klein j-invariant is $256(1-\lambda +{\lambda }^2{)}^3/[{\lambda }^2(1-\lambda {)}^2]$. Finally the same products giveIndeed ${\Theta }_2{\Theta }_3{\Theta }_4=2Q^{1/4}{\prod }_m(1-Q^{2m}{)}^3$, whose eighth power divided by $256$ is $q{\prod }_m(1-q^m{)}^{24}$. This recovers the normalized modular discriminant of Question 1 and displays the bridge between Gaussian theta sums and the lattice discriminant of Question 2.

A theta function with characteristics evaluated at spatial argument zero. Its dependence on $\tau \in \mathbb{H}$ remains holomorphic. Among the four half-integer characteristic choices, the odd one vanishes identically, whereas the three even constants are nonzero on the half-plane. The zeros of theta functions with characteristics locate their spatial zero away from the origin. Ratios of equal powers of even constants yield the modular lambda function and other modular-invariant functions.