Direct Siegel-style residue-calculus proofs of the modular inversion laws of Jacobi's theta functions $\vartheta_2$ and $\vartheta_4$

The Jacobi theta functions satisfy classical modular inversion laws under the generator $S:\tau\mapsto-1/\tau$ of the modular group. A modern proof program treats these laws by C. L. Siegel's residue-calculus method: Siegel (1954) proved the Dedekind eta inversion this way, Raji (2007) gave the analogous direct construction for $\vartheta_3$, and Mamah and Saraeb (Ramanujan J. 2024) gave it for $\vartheta_1$, deriving the $\vartheta_2$ and $\vartheta_4$ laws only indirectly from half-period identities. The open problem is to construct explicit meromorphic test functions and contour arguments in the same Siegel style that directly yield the inversion laws $\vartheta_2(z/\tau,-1/\tau)=(-i\tau)^{1/2}e^{\pi i z^2/\tau}\vartheta_4(z,\tau)$ and $\vartheta_4(z/\tau,-1/\tau)=(-i\tau)^{1/2}e^{\pi i z^2/\tau}\vartheta_2(z,\tau)$, with the Gaussian exponentials and multipliers emerging from the residue computation itself, without passing through the $\vartheta_1/\vartheta_3$ laws. No direct Siegel-style construction for these two functions appears in the cited literature, so the gap is open; the problem is methodological, since the laws themselves are classical.

Background

Let $\mathbb{H}=\{\tau\in\mathbb{C}:\operatorname{Im}\tau>0\}$ be the upper half-plane, fix $z\in\mathbb{C}$, and write $q=e^{\pi i\tau}$. The four Jacobi theta functions are the entire, quasi-periodic functions $\vartheta_1(z,\tau)=-i\sum_{n\in\mathbb{Z}}(-1)^n q^{(n+1/2)^2}e^{(2n+1)\pi i z},\quad \vartheta_2(z,\tau)=\sum_{n\in\mathbb{Z}} q^{(n+1/2)^2}e^{(2n+1)\pi i z},$ $\vartheta_3(z,\tau)=\sum_{n\in\mathbb{Z}} q^{n^2}e^{2n\pi i z},\quad \vartheta_4(z,\tau)=\sum_{n\in\mathbb{Z}}(-1)^n q^{n^2}e^{2n\pi i z},$ the standard building blocks of elliptic function theory. They transform under the modular group $\Gamma=SL_2(\mathbb{Z})$, generated by the inversion $S:\tau\mapsto-1/\tau$ and the translation $T:\tau\mapsto\tau+1$, as half-integral-weight objects with a Gaussian prefactor; all square roots below use the principal branch, $(-i\tau)^{1/2}=\exp(\tfrac12\log(-i\tau))$ with $-\pi<\arg\le\pi$. The behavior under $T$ is immediate from the series ($\vartheta_1,\vartheta_2$ pick up the eighth root of unity $e^{\pi i/4}$, while $\vartheta_3$ and $\vartheta_4$ are exchanged), so the substantive content is the inversion laws $\vartheta_1\!\left(\frac{z}{\tau},\frac{-1}{\tau}\right)=-i(-i\tau)^{1/2}e^{\pi i z^2/\tau}\vartheta_1(z,\tau),\qquad \vartheta_3\!\left(\frac{z}{\tau},\frac{-1}{\tau}\right)=(-i\tau)^{1/2}e^{\pi i z^2/\tau}\vartheta_3(z,\tau),$ $\vartheta_2\!\left(\frac{z}{\tau},\frac{-1}{\tau}\right)=(-i\tau)^{1/2}e^{\pi i z^2/\tau}\vartheta_4(z,\tau),\qquad \vartheta_4\!\left(\frac{z}{\tau},\frac{-1}{\tau}\right)=(-i\tau)^{1/2}e^{\pi i z^2/\tau}\vartheta_2(z,\tau).$ Note the structural feature that matters here: under $S$ the functions $\vartheta_1$ and $\vartheta_3$ are each mapped to a multiple of themselves, whereas $\vartheta_2$ and $\vartheta_4$ are exchanged; consequently $\vartheta_1$ transforms on the full group $\Gamma$ (with multiplier) while $\vartheta_2$ and $\vartheta_4$ are modular only for the congruence subgroup $\Gamma(2)$, the matrices congruent to the identity modulo 2. The zeros also differ: $\vartheta_1$ vanishes on the lattice $z=m+n\tau$, $\vartheta_2$ on its translate by $1/2$ (that is, $z=1/2+m+n\tau$), and $\vartheta_4$ on its translate by $\tau/2$ (that is, $z=\tau/2+m+n\tau$), for $m,n\in\mathbb{Z}$. Siegel (1954) proved the inversion transformation of the Dedekind eta function $\eta(\tau)=e^{\pi i\tau/12}\prod_{n\geq 1}(1-e^{2\pi i n\tau})$, the infinite product built from the same nome, by pure residue calculus: one expands a logarithm in a restricted domain, engineers a meromorphic test function whose residues reproduce the truncated expansion, integrates over an expanding contour, evaluates the boundary limits, and continues analytically. Rademacher later extended this to general modular matrices (as recounted in Raji's 2007 paper). Raji (2007) adapted the method to a Jacobi theta function for the first time, proving the $\vartheta_3$ inversion law directly. Mamah and Saraeb (2024) completed the analogous direct construction for $\vartheta_1$: for $\tau=iy$ with $z=a+ib$ restricted so that $b<0<a<1$ and $y>|b|$ (which makes all logarithmic series converge), they form the logarithmic difference $\phi(z,\tau)=\log\vartheta_1(z,\tau)-\log\vartheta_1(z/\tau,-1/\tau)$, expand it into geometric series in $q^{2m}$, $w^{2m}q^{2m}$, and $w^{-2m}q^{2m}$, and then build the explicit meromorphic test function $F_n(\zeta)=-\frac{1}{8\zeta}\cot(\pi i N\zeta)\cot\!\left(\pi N\frac{\zeta}{y}\right)+\frac{1}{\zeta}\left(\frac{1}{1-e^{2\pi N\zeta}}\right)\left(\frac{e^{-2\pi i(-\frac{zN}{y}+\frac{N}{y})\zeta}}{1-e^{-2\pi i\frac{N}{y}\zeta}}\right),\qquad N=n+\tfrac12,$ which has a pole of order 3 at $\zeta=0$ and simple poles at $\zeta=ik/N$ and $\zeta=ky/N$ for $k\in\mathbb{Z}\setminus\{0\}$. The residue theorem on a parallelogram contour $C$ with corners $\pm y,\pm i$ gives $2\pi i$ times a residue sum that reproduces the truncated series for $\phi$; the boundary limits $\lim_{n\to\infty}\zeta F_n(\zeta)=\pm1/8$ on the four edges, with the limit-interchange justified by the Arzela bounded convergence theorem, evaluate the integral as $-\log(y^{1/2})$; analytic continuation first in $\tau$ then in $z$ finishes the proof, the Gaussian factor and the multiplier $-i(-i\tau)^{1/2}$ emerging from the residue at the triple pole. For the remaining two functions the same paper does not give any construction: it notes the half-period relations $\vartheta_3(z+\tfrac12,\tau)=\vartheta_4(z,\tau)$ and $\vartheta_2(z+\tfrac12,\tau)=-\vartheta_1(z,\tau)$ (recorded in Rademacher's Topics in Analytic Number Theory, Chapter X) and asserts that similar results follow by a sequence of changes of variables. That derivation, however, routes the $\vartheta_2$ and $\vartheta_4$ laws through the already-proven $\vartheta_1$ and $\vartheta_3$ inversions combined with half-period and half-$\tau$ shift identities (the latter themselves carrying Gaussian prefactors); it is precisely this indirectness that the present problem asks to remove. Under $S$ the functions $\vartheta_2$ and $\vartheta_4$ are exchanged, and their zero lattices are shifted relative to those of $\vartheta_1$ and $\vartheta_3$.

Problem Statement

Construct direct residue-calculus proofs, in the manner of Siegel's proof for the Dedekind eta function as adapted by Raji for $\vartheta_3$ and by Mamah and Saraeb for $\vartheta_1$, of the modular inversion laws of the remaining two Jacobi theta functions.

Conventions (those of the source paper): $\tau\in\mathbb{H}$, $z\in\mathbb{C}$, $q=e^{\pi i\tau}$, $w=e^{\pi i z}$, and all powers use the principal branch with $-\pi<\arg\le\pi$. The two target identities are $\vartheta_2\!\left(\frac{z}{\tau},\frac{-1}{\tau}\right)=(-i\tau)^{1/2}\,e^{\pi i z^{2}/\tau}\,\vartheta_4(z,\tau),$ $\vartheta_4\!\left(\frac{z}{\tau},\frac{-1}{\tau}\right)=(-i\tau)^{1/2}\,e^{\pi i z^{2}/\tau}\,\vartheta_2(z,\tau).$

A valid answer must, for both $\vartheta_2$ and $\vartheta_4$: (i) exhibit an explicit meromorphic test function (or a family $F_n$ with $n\to\infty$, analogous to the $F_n$ used for $\vartheta_1$) together with its pole lattice and the domain of $z$ and $\tau$ in which the underlying logarithmic series converge; (ii) compute its residues exactly inside a stated expanding contour; (iii) prove the boundary-limit behavior on the contour and justify the interchange of limit and integral; (iv) carry out the analytic continuation from the initial domain (of the shape $\tau=iy$, $z$ restricted) to all of $\mathbb{H}\times\mathbb{C}$, threading around the zero lattices $z=1/2+m+n\tau$ of $\vartheta_2$ and $z=\tau/2+m+n\tau$ of $\vartheta_4$; and (v) obtain the Gaussian exponential $e^{\pi i z^{2}/\tau}$ and the multiplier $(-i\tau)^{1/2}$, including the root-of-unity prefactor (equal to 1 in these conventions, versus $-i$ for $\vartheta_1$), from the residue computation itself.

The common objective uniting these deliverables is to extend the Siegel-method program to the two theta functions it has not yet reached, and 'direct' is a binding requirement: the target identities must not be obtained by substituting into the already-proven $\vartheta_1$ or $\vartheta_3$ inversion laws, by combining those laws with the half-period or half-$\tau$ shift relations, or by translating or otherwise re-parameterizing the $\vartheta_1$ or $\vartheta_3$ test function; the test function for each of $\vartheta_2$ and $\vartheta_4$ must instead be built from that function's own logarithmic difference, $\log\vartheta_j(z,\tau)-\log\vartheta_j(z/\tau,-1/\tau)$ with $j\in\{2,4\}$. The five requirements (i)-(v) above are to be met jointly for each of the two functions. As in the source line, the behavior under the translation generator $T$ follows from the Fourier expansions and is not part of the task; the completion boundary is the pair of direct $S$-inversion proofs for $\vartheta_2$ and $\vartheta_4$.

References

1. Maher Mamah (listed as Maher Me'meh on arXiv) and Ali Saraeb, 'A generalization of Siegel's method to Jacobi's vartheta_1 function', The Ramanujan Journal 65 (2024), no. 2, 521-531. DOI: 10.1007/s11139-024-00894-7. arXiv:2208.02168. https://doi.org/10.1007/s11139-024-00894-7
2. Wissam Raji, 'A new proof of the transformation law of Jacobi's theta function theta_3(w,tau)', Proceedings of the American Mathematical Society 135 (2007), no. 10, 3127-3132. DOI: 10.1090/S0002-9939-07-08867-3. https://doi.org/10.1090/S0002-9939-07-08867-3
3. C. L. Siegel, 'A simple proof of eta(-1/tau) = eta(tau) sqrt(tau)/i', Mathematika 1 (1954), no. 1, 4. DOI: 10.1112/s0025579300000462. https://doi.org/10.1112/s0025579300000462
4. Maher Me'meh and Ali Saraeb, 'A Generalization of Iseki's Formula and The Jacobi Theta Function', Hardy-Ramanujan Journal 45 (2022), 130-139. DOI: 10.46298/hrj.2023.10912. https://doi.org/10.46298/hrj.2023.10912
5. Maher Me'meh and Ali Saraeb, 'A New Proof For The Transformation Laws of Theta Functions', preprint (2022). arXiv:2207.12806. https://arxiv.org/abs/2207.12806
6. Ali Saraeb and Maher Me'meh, 'On the Generalization of Radamacher's Proof to vartheta_1(z,tau)', preprint (2023). arXiv:2301.07561. https://arxiv.org/abs/2301.07561
7. NIST Digital Library of Mathematical Functions, Chapter 20 'Theta Functions', Section 20.7 'Identities', Subsection 20.7(viii) 'Transformations of Lattice Parameter', and Section 20.2 'Definitions and Periodic Properties', https://dlmf.nist.gov/20.7 and https://dlmf.nist.gov/20.2
8. Hans Rademacher, 'Topics in Analytic Number Theory', Grundlehren der mathematischen Wissenschaften 169, Springer-Verlag, 1973 (Chapter X, 'The vartheta-functions', pp. 164-183). DOI: 10.1007/978-3-642-80615-5. https://doi.org/10.1007/978-3-642-80615-5
9. Ze-Yong Kong and Lee Peng Teo, 'An Elementary Proof of the Transformation Formula for the Dedekind Eta Function', Armenian Journal of Mathematics 16 (2024), no. 4, 1-22. DOI: 10.52737/18291163-2024.16.4-1-22. https://doi.org/10.52737/18291163-2024.16.4-1-22