Liu's $q\to1$ degeneration conjecture: reduction of the $\chi_5$ theta-function identity (3.80) to the cyclotomic-unit logarithm

In his Hardy-Ramanujan Journal survey of a 'theta identities generating machine' based on a universal identity for theta functions of degree eight, Zhi-Guo Liu states as the paper's only conjecture that his identity (3.80) — an exact relation between the trigonometric quotient $\sin z\sin 2z/\sin 5z$, the $\chi_5$ Lambert sine series $\sum_{n\ge1}(n/5)q^n\sin 2nz/(1-q^n)$, and a quotient of Jacobi theta and Dedekind eta functions — 'reduces', as the elliptic nome $q\to1$, to the classical cyclotomic-unit formula $\sum_{n\ge1}(n/5)x^n/n=(1/\sqrt5)\log\big((2x^2+x+2+\sqrt5\,x)/(2x^2+x+2-\sqrt5\,x)\big)$. The target formula itself is elementary; the open problem is the degeneration statement: making precise in what sense the theta identity tends to the cyclotomic logarithm as the nome approaches the cusp, and proving it. The problem is open.

Background

Conventions. Throughout, $\tau$ lies in the complex upper half-plane, $q=e^{2\pi i\tau}$ is the elliptic nome ($|q|<1$), and $q\to1$ refers to approach of the nome to the cusp; the source states the conjectured limit only as 'When $q\to1$', without fixing the path of approach, any normalization, or the mode of convergence, so the precise limiting sense is itself part of the open problem. The Jacobi theta function $\theta_1(z|\tau)=2q^{1/8}\sin z\prod_{n\ge1}(1-q^n)(1-q^ne^{2iz})(1-q^ne^{-2iz})$ is odd and $\pi$-quasi-periodic in $z$; the Dedekind eta function is $\eta(\tau)=q^{1/24}\prod_{n\ge1}(1-q^n)$. A Lambert series is a series of the form $\sum_{n\ge1}a_nq^n/(1-q^n)$. Here $(n/5)$ denotes the Legendre symbol modulo 5, i.e. the quadratic Dirichlet character $\chi_5$ with $\chi_5(n)=0$ if $5\mid n$ and $\chi_5(n)=\pm1$ according as $n\equiv\pm1$ or $\pm2\pmod5$.

The generating theorem. Liu's Theorem 1.11 (A universal identity for theta functions of degree eight and applications, Hardy-Ramanujan J. 43 (2020) 129-172) states that any even entire function $f(z|\tau)$ with the Weierstrass-sigma quasi-periodicity $f(z|\tau)=f(z+\pi|\tau)=q^4e^{16iz}f(z+\pi\tau|\tau)$ satisfies a fixed linear identity of degree eight in theta functions, relating the quotient $f(x|\tau)/\theta_1^2(2x|\tau)$ to the analogous quotient at a second arbitrary point $y$, with coefficients built from the values of $f$ at the four half-period points $0,\pi/2,(\pi+\pi\tau)/2,\pi\tau/2$; Theorem 1.12, obtained from Theorem 1.11 by asymptotic analysis, is a companion identity relating the derivatives $(\log f)''(0|\tau)$ and $(\log f)^{(4)}(0|\tau)$ and the Eisenstein series $E_2(\tau),E_4(\tau)$ (standard modular forms expressible as Lambert series) to those same half-period values of $f$. Specializing $f$ recovers classical identities of Jacobi, Kiepert (the quintuple product identity), Ramanujan, and Weierstrass, whence the name 'theta identities generating machine'.

The specific identity. Applying the machine to a suitable auxiliary function (in the published survey the quasi-periodicity conditions are verified for the quotient built from the left-hand side below), Liu derives the identity, first established as Proposition 5.1 of his J. Number Theory 132 (2012) 2955-2966 paper and restated as Eq. (3.80) of the Hardy-Ramanujan Journal paper: for $\mathrm{Im}\,\tau>0$ and $z$ away from the zeros of $\sin 5z$, $\frac{\sin z\sin 2z}{\sin 5z}-\sum_{n\ge1}\left(\frac n5\right)\frac{q^n}{1-q^n}\sin 2nz=\frac{\eta^2(5\tau)\,\theta_1(z|\tau)\theta_1(2z|\tau)}{2\,\eta(\tau)\,\theta_1(5z|5\tau)}.$ The left side is real for real $z$ and real $q$; the companion identity (3.81), a Lambert series with denominator $1-q^{5n}$, is implied in Theorem 1 of his J. Ramanujan Math. Soc. 22 (2007) 283-298 paper on the Eisenstein series of $\Gamma_0(5)$ (the modular group of integer matrices with lower-left entry divisible by 5).

The target formula. The conjectured limiting form is the classical cyclotomic-unit identity, for $|x|<1$: $\sum_{n\ge1}\left(\frac n5\right)\frac{x^n}{n}=\frac{1}{\sqrt5}\log\frac{2x^2+x+2+\sqrt5\,x}{2x^2+x+2-\sqrt5\,x}$. This identity is elementary: differentiating both sides gives the same rational function $\frac{1-x^2}{1+x+x^2+x^3+x^4}$ and both sides vanish at $x=0$; equivalently $2x^2+x+2+\sqrt5\,x=2(x-e^{4\pi i/5})(x-e^{-4\pi i/5})$ and $2x^2+x+2-\sqrt5\,x=2(x-e^{2\pi i/5})(x-e^{-2\pi i/5})$ factor over the primitive 5th roots of unity, and $x\to1^-$ recovers the classical evaluation of the Dirichlet $L$-value $L(1,\chi_5)$ (the value at $s=1$ of the Dirichlet series $\sum_{n\ge1}\chi_5(n)n^{-s}$), namely $L(1,\chi_5)=\frac{2}{\sqrt5}\log\frac{1+\sqrt5}{2}$. The substantive content of the conjecture is therefore not the target formula itself but the degeneration statement connecting it to (3.80).

Structure of the limit (numerical observations). As $q\to1^-$, the trigonometric prefactor $\sin z\sin 2z/\sin 5z$ is $q$-independent, while the Lambert series is singular on some intervals of $z$ and both sides of (3.80) are exponentially small on the complementary intervals, so that the pointwise limit of the unscaled identity is piecewise degenerate; some normalization is therefore required to extract a nontrivial limit. General machinery for the Lambert-series side (complete asymptotic expansions of Lambert series near $q=1$) is available in the work of Banerjee and Wilkerson, but has not been applied to this identity.

Problem Statement

Prove or refute Liu's Conjecture 3.10 (Hardy-Ramanujan J. 43 (2020) 129-172; arXiv:2104.14705): as $q\to1$, the theta-function identity $\frac{\sin z\sin 2z}{\sin 5z}-\sum_{n\ge1}\left(\frac n5\right)\frac{q^n}{1-q^n}\sin 2nz=\frac{\eta^2(5\tau)\,\theta_1(z|\tau)\theta_1(2z|\tau)}{2\,\eta(\tau)\,\theta_1(5z|5\tau)}$ degenerates to the cyclotomic-unit formula $\sum_{n\ge1}\left(\frac n5\right)\frac{x^n}{n}=\frac{1}{\sqrt5}\log\frac{2x^2+x+2+\sqrt5\,x}{2x^2+x+2-\sqrt5\,x},\qquad |x|<1.$ The source states the conjecture in a single sentence — 'When $q\to1$, the identity in (3.80) reduces to the formula (3.83)' — and specifies neither the path of approach to the cusp, nor any normalization, nor the mode of convergence in $z$, nor the treatment of the exceptional points $z=k\pi/5$ (the zeros of $\sin 5z$), nor the branch of the logarithm, nor whether an analytic or a formal/termwise reading of 'reduces to' is intended. Fixing the sense of the reduction is therefore part of the open problem: a resolution must first adjudicate the conjecture's literal sentence — whether it holds as stated — and then either state precisely the sense in which the degeneration holds and establish it rigorously, or exhibit explicitly where the conjecture as literally stated fails, together with the corrected statement. Nothing in the Background prescribes a required proof route or a required intermediate conclusion. The single scientific objective is to settle Conjecture 3.10 — the only stated conjecture of Liu's degree-eight theta-identity machine — by rigorously establishing or refuting the degeneration of the identity (3.80) to the classical cyclotomic-unit logarithm at the cusp.

References

1. Zhi-Guo Liu, A universal identity for theta functions of degree eight and applications, Hardy-Ramanujan Journal 43 (2020) 129-172, DOI:10.46298/hrj.2021.7432, arXiv:2104.14705, https://arxiv.org/abs/2104.14705
2. Zhi-Guo Liu, A theta function identity of degree eight and Eisenstein series identities, Journal of Number Theory 132 (2012) 2955-2966, DOI:10.1016/j.jnt.2012.06.007, https://doi.org/10.1016/j.jnt.2012.06.007
3. Zhi-Guo Liu, A theta function identity and the Eisenstein series on Γ_0(5), Journal of the Ramanujan Mathematical Society 22 (2007), No. 3, 283-298, zbMATH:5269411, https://zbmath.org/5269411
4. Zhi-Guo Liu, The Kronecker theta function and a decomposition theorem for theta functions I, The Ramanujan Journal 61 (2021) 149-173, DOI:10.1007/s11139-020-00376-6, https://doi.org/10.1007/s11139-020-00376-6
5. Shubho Banerjee, Blake Wilkerson, Asymptotic expansions of Lambert series and related q-series, International Journal of Number Theory 13 (2017) 2097-2113, DOI:10.1142/S1793042117501135, https://doi.org/10.1142/S1793042117501135