Jacobi theta function ($\theta (\tau )$)

The Jacobi theta functionsatisfies $\theta (\tau +2)=\theta (\tau )$ and $\theta (-1/\tau )=(-i\tau {)}^{1/2}\theta (\tau )$.

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 real characteristics $\alpha ,\beta$ and $\mathrm{Im}\tau >0$, Gaussian decay gives normal convergence, an entire function of $z$ and holomorphic dependence on $\tau$. Integer shifts of alpha leave the function unchanged; a unit shift of beta multiplies it by $e^{2\pi i\alpha }$. Its spatial quasi-periods are $\theta (z+1)=e^{2\pi i\alpha }\theta (z)$ and $\theta (z+\tau )=e^{-\pi i\tau -2\pi i(z+\beta )}\theta (z)$. Binary characteristic labels often mean alpha and beta divided by two, so the convention must be stated.

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.

Use the radian spatial convention ${\theta }_3(z,\tau )={\sum }_n{e}^{\pi i{n}^2\tau +2inz}$ and ${\theta }_4(z,\tau )={\theta }_3(z+\pi /2,\tau )$. Their ratio is unchanged by $\pi$ and changes sign under $\pi \tau$. The numerator has simple zeros at $\pi (1+\tau )/2+\pi \mathbb{Z}+\pi \tau \mathbb{Z}$, while the denominator has simple zeros at $\pi \tau /2+\pi \mathbb{Z}+\pi \tau \mathbb{Z}$. The two sets are disjoint, so any period must lie in $\pi \mathbb{Z}+\pi \tau \mathbb{Z}$, by translating the zero set. The sign rule then gives the displayed exact lattice. Squaring the ratio removes that sign and has precisely the smaller lattice $\pi \mathbb{Z}+\pi \tau \mathbb{Z}$.

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.

Translation of tau by one gives $\theta [\alpha ,\beta ](z,\tau +1)=e^{-\pi i\alpha (\alpha -1)}\theta [\alpha ,\beta +\alpha -1/2](z,\tau )$. Gaussian Poisson summation gives $\theta [\alpha ,\beta ](z/\tau ,-1/\tau )=\sqrt{-i\tau }{e}^{\pi i{z}^2/\tau +2\pi i\alpha \beta }\theta [\beta ,-\alpha ](z,\tau )$. The square root is the holomorphic branch positive at tau=i. These formulas keep the characteristic phase, which is lost if one states only the transformation of unshifted theta constants.

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.

The logarithmic derivative is periodic under one and decreases by $2\pi i$ under tau. Integrating around a fundamental cell and applying the argument principle counts exactly one zero, with multiplicity. The displayed zero follows from translating the zero of $\theta [0,0]$ at $(1+\tau )/2$. It is consequently simple. For half-integer characteristics only the odd characteristic $(1/2,1/2)$ vanishes at the origin; the other three theta constants are nonzero.

The identity between Jacobi theta functions has the oscillator-product form ${\prod }_{r\in \mathbb{N}-1/2}(1+q^r{)}^8-{\prod }_{r\in \mathbb{N}-1/2}(1-q^r{)}^8=16q^{1/2}{\prod }_{n\ge 1}(1+q^n{)}^8$ for $|q|<1$. In a critical RNS string, dividing this identity by the eight-boson oscillator product proves equality of GSO projection-retained Neveu–Schwarz sector and Ramond sector state multiplicities at every mass level.

In the $e^{\pi iz}$ convention, Poisson summation gives ${\Theta }_3(-1/z)=\sqrt{-iz}{\Theta }_3(z)$ and interchanges ${\Theta }_2,{\Theta }_4$ with the same factor. Translation by one sends ${\Theta }_2$ to $e^{\pi i/4}{\Theta }_2$ and interchanges ${\Theta }_3,{\Theta }_4$. Eighth powers remove every phase and square-root ambiguity when proving integral-weight modularity.

This convention for the Jacobi theta function is the usual theta constant evaluated at twice the argument. The Poisson summation formula gives $\theta (-1/(4\tau ))=\sqrt{-2i\tau }\theta (\tau )$, with the square root holomorphic on the half-plane and positive on the imaginary axis after evaluating its positive real argument.

The number of ordered integer eight-tuples with square sum $n$ satisfies the displayed divisor formula for positive $n$, with $r_8(0)=1$ separately. The eighth power of the shifted theta series of integer squares is $(16E_4(2\tau )-E_4(\tau ))/15$. Its coefficient sign is $(-1{)}^n$, because squares and their integer roots have the same parity.

For $|q|<1$ and $z\ne 0$, the Jacobi triple product is

The odd Dirichlet character modulo four has ${\chi }_4(2r+1)=(-1{)}^r$. The series $B=\frac{1}{2}{\sum }_{n}n{\chi }_4(n)q^{n^2/8}$ is therefore $q^{1/8}$ times the displayed sum. The character theta proof of the eta product gives $B={\eta }^3$, proving the identity.

The identity holds for $|q|<1$, with both sides locally uniformly convergent. In the character theta proof of the eta product, pair the terms indexed by $n$ and $-n$ and then select $n=6r+1$; their Dirichlet character signs are $(-1{)}^r$. Removing $q^{1/24}$ gives the displayed product identity.

The theta group is ${\Gamma }_{\theta }=\Gamma (2)\cup \Gamma (2)S$. It is generated by $T^2$ and $S$ and has index three in the modular group.

The Jacobi triple product givesEvery factor is nonzero for $|q|<1$, and the product converges to a nonzero limit, so $\theta$ has no zero in the upper half-plane.