Liu's Conjecture 3.10: the $q\to1$ degeneration of a degree-eight theta-function identity to a quadratic-character logarithmic series

Liu's universal identity for theta functions of degree eight (Hardy-Ramanujan Journal 43, 2021) generates hundreds of classical theta-function identities, among them an identity (3.80) equating a trigonometric rational function minus a Lambert series in the quadratic character $(n/5)$ with an explicit quotient of Dedekind eta and Jacobi theta functions. The paper's only conjecture (Conjecture 3.10) states that as $q\to1$ this identity 'reduces to' the closed-form series $\sum_{n\ge1}(n/5)x^n/n=\frac{1}{\sqrt5}\log\frac{2x^2+x+2+\sqrt5x}{2x^2+x+2-\sqrt5x}$. The target formula is elementary — a partial-fractions consequence of the discrete Fourier expansion of the character mod 5 — so the open content is the degeneration claim itself: making the singular $q\to1$ limit precise (the two sides' behaviour as $q\to1$ is non-uniform in $z$, so a renormalization and an identification of the variable $x$ must be derived) and proving the resulting asymptotic theorem for both sides.

Background

Liu's 2021 paper establishes a 'universal' identity for theta functions of degree eight and demonstrates that it acts as a generating machine for classical theta-function identities. Three standard objects are needed. The Jacobi theta function $\theta_1(z|\tau)$ is the odd theta function defined, for the nome $q=e^{2\pi i\tau}$ with $|q|<1$, by the product $\theta_1(z|\tau)=2q^{1/8}\sin z\prod_{n\ge1}(1-q^n)(1-q^ne^{2iz})(1-q^ne^{-2iz})$; 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 $\sum_{n\ge1}a_n\frac{q^n}{1-q^n}$. Finally, $(n/5)$ denotes the Legendre symbol, i.e. the quadratic Dirichlet character modulo 5, taking values $1,-1,-1,1,0$ for $n\equiv1,2,3,4,0\pmod 5$.

Liu's Theorem 1.11 states that any even entire function $f(z|\tau)$ satisfying the quasi-periodicity conditions $f(z|\tau)=f(z+\pi|\tau)=q^4e^{16iz}f(z+\pi\tau|\tau)$ (a function 'of degree eight') obeys a universal identity relating its values at two points to its values at the four points $0$, $\pi/2$, $(\pi+\pi\tau)/2$, and $\pi\tau/2$ (the origin and the three half-periods); it is a natural extension of the addition formula for the Weierstrass sigma function, the classical odd quasi-periodic entire function attached to a lattice. Hundreds of identities for elliptic modular functions, classical and new, are corollaries. One such corollary — Theorem 3.9 of the paper, first established as Proposition 5.1 of Liu's 2012 Journal of Number Theory paper — is proved by applying Theorem 3.1, a degree-four theta-function identity derived from Theorem 1.11, and reads

$\frac{\sin z\,\sin 2z}{\sin 5z}-\sum_{n=1}^{\infty}\left(\frac{n}{5}\right)\frac{q^n}{1-q^n}\sin 2nz=\frac{\eta^2(5\tau)\,\theta_1(z|\tau)\theta_1(2z|\tau)}{2\eta(\tau)\theta_1(5z|5\tau)}.$

Theta identities have two classical degeneration limits. The limit $q\to0$ is regular ($\theta_1\to2q^{1/8}\sin z$, $\eta\to q^{1/24}$) and yields trigonometric identities; Liu uses it elsewhere in the paper. The opposite limit $q\to1$ (equivalently $\tau\to0$; write $q=e^{-t}$ with $t\to0^+$) is singular, and its behaviour is non-uniform in $z$: at the term level $\frac{q^n}{1-q^n}\sim\frac{1}{nt}$, so the Lambert series on the left can grow like $\frac{1}{t}$ for suitable $z$ while remaining bounded for other $z$, and the eta-theta quotient on the right is governed by the imaginary transformation of theta functions (the modular substitution $\tau\mapsto-1/\tau$) and the modularity of $\eta$; the displayed identity degenerates at the points $z\equiv\pi k/5$. Liu nevertheless conjectured (Conjecture 3.10, the paper's only conjecture) that when $q\to1$ the identity above 'reduces to' the closed formula

$\sum_{n=1}^{\infty}\left(\frac{n}{5}\right)\frac{x^n}{n}=\frac{1}{\sqrt5}\log\frac{2x^2+x+2+\sqrt5\,x}{2x^2+x+2-\sqrt5\,x}$

whose left side is the integrated generating function of the quadratic character modulo 5 and whose $x\to1$ specialization is the classical Dirichlet $L$-value evaluation $L(1,\chi_5)=\frac{2}{\sqrt5}\log\frac{1+\sqrt5}{2}$, where $L(s,\chi_5)=\sum_{n\ge1}\chi_5(n)n^{-s}$ is the Dirichlet $L$-function of the same character $\chi_5(n)=(n/5)$. The conjecture is stated in one sentence: no renormalization, no identification of the new variable $x$ in terms of $z$ or $q$, and no limiting domain are specified, so making the singular limit precise is itself part of the mathematical content.

Problem Statement

Prove or refute Liu's Conjecture 3.10 (Hardy-Ramanujan Journal 43, 2021): establish, in a precise and justified sense, that the $q\to1$ degeneration of the theta identity (3.80) yields the logarithmic identity (3.83). The single objective is a rigorous degeneration theorem connecting (3.80) to (3.83); a complete answer must jointly provide:

1. A precise operationalization of the limit: the renormalization under which both sides of (3.80) have finite limits as $q\to1$ (for example an explicit power of $1-q$ or of $-\log q$), the identification of the variable $x$ of (3.83) in terms of $z$ (and of $q$ as well, if a boundary-layer scaling of $z$ with $q$ is used), and the parameter domain (real $z$ versus complex $z$ in the fundamental parallelogram), with justification that this reading captures the conjecture.

2. A proof of the degeneration of both sides of (3.80) under that renormalization: the Lambert-series side together with its trigonometric term, and the eta-theta-quotient side, with matching limits, explicit error bounds, and control at the singular configurations of the limit — the points $z\equiv\pi k/5$, which are at once the zeros of $\sin 5z$ and the jump points of the limiting piecewise-constant function.

3. The derivation of formula (3.83) as the exact limiting identity, including the correct branch of the logarithm on the stated domain, recovering the classical evaluation $L(1,\chi_5)=\frac{2}{\sqrt5}\log\frac{1+\sqrt5}{2}$ at $x\to1$.

A refutation must instead exhibit a precise, natural operationalization of 'the identity reduces to' under which the claimed reduction fails (for instance, the renormalized limit exists but differs from (3.83), or no such limit exists), together with a rigorous proof or an exact computation of the failure; a reading that merely lets both sides diverge, without extracting the finite part that the word 'reduces' promises, is not a fair reading. The Appell-Lerch extension avenue mentioned in the paper's conclusion is a separate open-ended direction, not a required deliverable of this problem.

References

1. Zhi-Guo Liu, A universal identity for theta functions of degree eight and applications, Hardy-Ramanujan Journal 43 (2021), 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, On a Transformation of Triple $q$-Series and Rogers-Hecke Type Series, SIGMA 20 (2024), 086, DOI: 10.3842/SIGMA.2024.086, arXiv:2401.14099, https://arxiv.org/abs/2401.14099
4. Heng Huat Chan, Song Heng Chan, Zhi-Guo Liu, Wadim Zudilin, On Schultz's generalization of Borweins' cubic identity, Advances in Mathematics 495 (2026), article 110967, DOI: 10.1016/j.aim.2026.110967, arXiv:2511.15519, https://arxiv.org/abs/2511.15519