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Dedekind eta function

The Dedekind eta function, denoted \eta(\tau), is a holomorphic cusp form of weight $1/2 defined on the upper half-plane \mathbb{H} = \{\tau \in \mathbb{C} \mid \Im(\tau) > 0\} by the infinite product formula

\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1 - q^n),

where q = e^{2\pi i \tau}.^[1] Introduced by the German mathematician Richard Dedekind in 1877 as part of his investigations into elliptic modular functions, it serves as a fundamental building block in the theory of modular forms and algebraic number theory.^[2]^[1] The eta function exhibits nontrivial transformation properties under the action of the modular group \mathrm{SL}(2, \mathbb{Z}). Specifically, for \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2, \mathbb{Z}) with c > 0,

\eta\left( \frac{a\tau + b}{c\tau + d} \right) = \epsilon(a, b, c, d) \, (-i (c\tau + d))^{1/2} \, \eta(\tau),

where \epsilon(a, b, c, d) is a 24th root of unity involving a Dedekind sum s(-d, c).^[1] A key special case is the inversion formula \eta(-1/\tau) = (-i\tau)^{1/2} \eta(\tau).^[1] Raising the eta function to the 24th power yields the modular discriminant

\Delta(\tau) = \eta(\tau)^{24} = (2\pi)^{12} q \prod_{n=1}^\infty (1 - q^n)^{24},

which is a cusp form of weight 12 that generates (together with the Eisenstein series E_4 and E_6) the ring of all modular forms for \mathrm{SL}(2, \mathbb{Z}).^[3] Beyond its structural role in modular forms, the Dedekind eta function has profound connections to analytic number theory. Its reciprocal provides the generating function for the partition numbers p(n):

\frac{1}{\eta(\tau)} = q^{-1/24} \sum_{n=0}^\infty p(n) q^n,

a relation exploited by Hardy and Ramanujan in their 1918 circle method derivation of the asymptotic formula for p(n).^[3] Eta-quotients—finite products of the form \prod_{\delta} \eta(\delta \tau)^{r_\delta} for integers r_\delta—span important subspaces of modular forms for congruence subgroups and appear in explicit formulas for arithmetic invariants, such as L-values and class numbers of quadratic fields.^[3] These properties underscore the eta function's enduring significance in bridging analysis, algebra, and combinatorics.^[1]

Definition and Fundamentals

Infinite Product Representation

The Dedekind eta function \eta(\tau) is defined for \tau in the upper half-plane \mathbb{H} = \{ \tau \in \mathbb{C} \mid \Im(\tau) > 0 \} by the infinite product representation

\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1 - q^n),

where q = e^{2\pi i \tau} satisfies |q| < 1, ensuring the product's absolute convergence. This form arises as a q-analog of the Euler function, with the factor q^{1/24} providing normalization for modular properties.^[4] The domain \mathbb{H} is essential, as \Im(\tau) > 0 guarantees |q| < 1, which is required for the infinite product to converge pointwise; moreover, the convergence is uniform on any compact subset of \mathbb{H}. Consequently, \eta(\tau) defines a holomorphic function on the entire upper half-plane \mathbb{H}, with no zeros or poles in this region. As a q-series, \eta(\tau) admits a Fourier expansion via Euler's pentagonal number theorem:

\eta(\tau) = \sum_{k=-\infty}^\infty (-1)^k q^{(3k^2 - k)/2 + 1/24},

where the exponents (3k^2 - k)/2 are generalized pentagonal numbers, and the coefficients \pm 1 (with sign (-1)^k) relate inversely to the partition function p(n), since the reciprocal $1/\eta(\tau) generates the partitions as \sum_{n=0}^\infty p(n) q^n.^[4] The exponent $1/24 originates from the transformation behavior under integer translations, where the unnormalized product acquires a phase e^{2\pi i / 24}, and is tied to the Ramanujan constant through the 24th power \eta(\tau)^{24} forming the modular discriminant.^[5]

Relation to Modular Discriminant

The modular discriminant, denoted \Delta(\tau), is defined as

\Delta(\tau) = \eta(\tau)^{24},

a cusp form of weight 12 for the modular group \mathrm{SL}(2, \mathbb{Z}).^[5] The space of cusp forms of weight 12 for \mathrm{SL}(2, \mathbb{Z}) is one-dimensional, so \Delta(\tau) is the unique normalized cusp form in this space, with leading Fourier coefficient 1.^[5] The q-expansion of \Delta(\tau) takes the form

\Delta(\tau) = q \prod_{n=1}^\infty (1 - q^n)^{24} = \sum_{n=1}^\infty \tau(n) q^n,

where q = e^{2\pi i \tau} and \tau(n) denotes the values of the Ramanujan tau function.^[5] Since the Dedekind eta function transforms as a modular form of weight $1/2, raising it to the 24th power yields \eta(\tau)^{24} as a modular form of integer weight 12, which aligns with the properties of \Delta(\tau).^[6]

Modular Properties

Transformation Laws

The transformation laws of the Dedekind eta function \eta(\tau) under the action of the modular group \mathrm{SL}(2, \mathbb{Z}) are fundamental to its role as a modular object of weight $1/2. These laws express how \eta(\tau) transforms when \tau in the upper half-plane is replaced by \gamma \tau = \frac{a\tau + b}{c\tau + d} for \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2, \mathbb{Z}) with integer entries and determinant 1. The group is generated by the matrices T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} (translation) and S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} (inversion), and the laws are first established for these generators before extending to the full group.^[5] Under the translation T: \tau \mapsto \tau + 1,

\eta(\tau + 1) = e^{\pi i / 12} \eta(\tau),

where e^{\pi i / 12} is a primitive 24th root of unity.^[7] This phase shift arises from the leading q^{1/24} term in the product's expansion, as the nome q = e^{2\pi i \tau} transforms to q e^{2\pi i}, introducing the factor e^{\pi i / 12}.^[5] Under the inversion S: \tau \mapsto -1/\tau,

\eta(-1/\tau) = \sqrt{-i \tau} \, \eta(\tau),

where the square root denotes the principal branch with argument in (-\pi/2, \pi/2].^[7] This reflects the half-integral weight, with the factor \sqrt{-i \tau} ensuring consistency across the modular transformations.^[8] For a general \gamma \in \mathrm{SL}(2, \mathbb{Z}) with c > 0, the transformation is given by

\eta(\gamma \tau) = \varepsilon(\gamma) (-i (c \tau + d))^{1/2} \eta(\tau),

where \varepsilon(\gamma) is a 24th root of unity depending on \gamma, and the square root is again the principal branch.^[7]^[5] The map \gamma \mapsto \varepsilon(\gamma) forms a homomorphism from \mathrm{SL}(2, \mathbb{Z}) to the multiplicative group of 24th roots of unity, ensuring the overall transformation factor multiplies compatibly under group composition.^[8] These laws can be derived from the infinite product representation \eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1 - q^n) by substituting the transformed nome q' = e^{2\pi i (a\tau + b)/(c\tau + d)}. The resulting product is analyzed using the Jacobi triple product identity, which relates it to a theta series, followed by application of the Poisson summation formula to obtain the modular behavior; induction on c (assuming c > 0) incorporates Dedekind sums to yield the explicit \varepsilon(\gamma).^[7] This approach confirms the holomorphy and the precise phase factors without relying on elliptic function theory.^[5]

Weight and Multiplier System

The Dedekind eta function \eta(\tau) is a modular form of weight $1/2 on the full modular group \mathrm{SL}_2(\mathbb{Z}), transforming under the action of \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}) as \eta(\gamma \tau) = v_\eta(\gamma) \eta(\tau), where the multiplier system is given by v_\eta(\gamma) = \varepsilon(\gamma) (-i (c\tau + d))^{1/2} for c > 0. Here, \varepsilon(\gamma) is a 24th root of unity depending on the entries of \gamma and expressed in terms of Dedekind sums s(-d, c).^[5]^[9] The factor \varepsilon(\gamma) incorporates a Dirichlet character aspect through its dependence on d \mod 24, related to the Kronecker symbol \left( \frac{d}{24} \right), which ensures consistency across the group action; specifically, for the inversion transformation, it aligns with the principal branch of the square root. The half-integral weight introduces non-holomorphic behavior via the (c\tau + d)^{1/2} factor, where the square root is defined using the principal branch with argument in (-\pi, \pi), chosen such that \operatorname{Re}((c\tau + d)^{1/2}) > 0 for \tau in the upper half-plane, guaranteeing the transformation law holds without singularities.^[5] This multiplier system distinguishes \eta(\tau) within the theory of half-integral weight modular forms. The Shimura correspondence provides a map from spaces of weight k + 1/2 forms with such multipliers to integer weight $2k + 2 forms, lifting \eta(\tau) and its twists to cusp forms like powers related to the modular discriminant \Delta(\tau) = \eta(\tau)^{24}. The space of modular forms of weight $1/2 on \mathrm{SL}_2(\mathbb{Z}) with the eta multiplier system is one-dimensional, uniquely determining \eta(\tau) up to a nonzero scalar multiple.^[10]

Identities and Connections

Combinatorial Identities

The Euler function \phi(q) = \prod_{n=1}^\infty (1 - q^n), with q = e^{2\pi i \tau}, is directly related to the Dedekind eta function by \phi(q) = q^{-1/24} \eta(\tau). Euler's pentagonal number theorem expresses this product as a bilateral series:

\phi(q) = \sum_{k=-\infty}^\infty (-1)^k q^{k(3k-1)/2},

where the exponents k(3k-1)/2 for integer k are the generalized pentagonal numbers (including both positive and negative indices).^[11] This identity, first proved by Leonhard Euler through iterative expansion and pairing of terms in the infinite product, provides a combinatorial interpretation: the coefficients are zero except at pentagonal numbers, where they alternate in sign as \pm 1, corresponding to restricted partitions with signs based on the parity of the number of summands in certain distinct-part representations.^[11] The theorem's reciprocal form, $1/\phi(q) = \sum_{n=0}^\infty p(n) q^n, generates the unrestricted partition function p(n), enabling a recurrence relation for p(n) via the pentagonal coefficients, which subtracts partitions aligned with pentagonal indices.^[12] Michael Somos discovered thousands of combinatorial identities involving products of Dedekind eta functions at rational multiples of \tau, often of finite level N where arguments are scaled by fractions with denominator N. A representative level-6 identity is

\eta(\tau)^2 \eta(2\tau)^2 \eta(3\tau)^2 \eta(6\tau)^2 = \eta(\tau/2)^2 \eta(\tau/3)^2 \eta(2\tau/3)^2 \eta(3\tau/2)^2,

equating products over the sixfold scaling group.^[13] Such identities arise from computational enumeration using q-series expansions and can be proved via modular transformations or Rogers-Ramanujan type dissections, where the eta products are expanded and coefficients matched term-by-term after applying valence formulas to ensure modular invariance.^[13] Srinivasa Ramanujan independently derived numerous eta product identities in his notebooks, often linking them to partition congruences or continued fractions, predating Somos's computational approach.^[14] For instance, one such relation expresses products like \eta(\tau) \eta(4\tau) \eta(5\tau) \eta(20\tau) in terms of powers of eta at scaled arguments, such as \eta(2\tau)^3 \eta(10\tau)^3, reflecting cubic relations from class number constraints.^[14] Proofs typically involve q-series manipulations, such as substituting the product form of eta and applying identities like the Jacobi triple product to regroup terms, or leveraging modular equations of degree matching the level to verify equality under the full modular group.^[14] These combinatorial identities highlight the eta function's role in bridging infinite products to finite sums over partition-like structures, with applications in enumerative combinatorics.

Links to Theta and Euler Functions

The Dedekind eta function exhibits deep analytic connections to Jacobi theta functions, which are fundamental in the theory of elliptic functions. These identities highlight how the eta function, defined via an infinite product, can be recast as a theta series, facilitating proofs of modular properties through the well-known transformation behaviors of theta functions. The eta function is also intimately tied to the Euler function \phi(q) = \prod_{n=1}^\infty (1 - q^n), a q-series that generates partition identities via its reciprocal and expansions. Specifically, \phi(q) = q^{-1/24} \eta(\tau), establishing the eta as a normalized version of the Euler function with modular weight $1/2. This relation underpins applications in partition theory, where the eta function's coefficients relate to the generating function for the number of partitions into distinct parts, adjusted by the q^{1/24} prefactor. Jacobi's triple product identity provides a crucial bridge between theta functions and products resembling the eta function: \prod_{n=1}^\infty (1 - q^{2n}) (1 + z q^{2n-1}) (1 + q^{2n-1}/z) = \sum_{m=-\infty}^\infty q^{m^2} z^{2m}, which specializes to theta series and connects to the Euler function through limiting cases where z = 1.^[15] By setting appropriate values, this identity yields product representations for individual theta functions, from which eta-quotients emerge as ratios, illustrating the eta's role in unifying product and sum forms in q-series.^[16] Relations between eta and theta functions are often derived using the Poisson summation formula, which equates \sum_{n \in \mathbb{Z}} f(n) to \sum_{k \in \mathbb{Z}} \hat{f}(k) for suitable test functions f, applied to Gaussian sums defining theta series. For instance, applying Poisson summation to the theta kernel e^{-\pi n^2 t} yields the modular transformation \theta(1/t) = \sqrt{t} \theta(t), from which the eta function's inversion formula \eta(-1/\tau) = \sqrt{-i\tau} \eta(\tau) follows via differentiation or logarithmic relations.^[17] This Fourier-analytic approach underscores the eta-theta interplay in establishing modularity. These links trace back to Carl Gustav Jacob Jacobi's foundational work in the 1820s and 1830s, where he developed the theory of theta functions and their product identities well before Richard Dedekind formalized the eta function in 1877, providing the analytic groundwork for later modular form developments.^[18]

Special Values

Evaluations at Specific Points

The Dedekind eta function can be evaluated explicitly at specific quadratic irrational points in the upper half-plane, yielding closed-form expressions involving the gamma function. These evaluations are derived from connections to elliptic integrals and the reflection formula for the gamma function, providing key insights into the function's behavior at fixed points. At \tau = i, the value is given by

\eta(i) = \frac{\Gamma\left(\frac{1}{4}\right)}{2 \pi^{3/4}}.

This expression arises from the relation between the eta function and the complete elliptic integral of the first kind, with the gamma function appearing via the beta function identity \mathrm{B}(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} \, dt = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}.^[6] At \tau = \rho = e^{2\pi i / 3}, a primitive cube root of unity with positive imaginary part, the evaluation is

\eta(\rho) = e^{-i \pi / 24} \, 3^{1/8} \, \frac{\Gamma\left(\frac{1}{3}\right)^{3/2}}{2\pi}.

This formula similarly stems from integral representations linking the eta function to periods of elliptic curves over the Eisenstein integers, incorporating the gamma function through multiple gamma generalizations or Chowla-Selberg-type products reduced for class number 1 fields.^[19] At cusps, the eta function vanishes in the limit as the imaginary part of \tau tends to infinity: \eta(\infty) = 0, consistent with the q-expansion where q^{1/24} \prod_{n=1}^\infty (1 - q^n) \to 0 as q = e^{2\pi i \tau} \to 0. For rational cusps, such as \tau = r/s in lowest terms, the value is defined via the modular transformation law \eta\left(\frac{a\tau + b}{c\tau + d}\right) = \epsilon \, (-i (c\tau + d))^{1/2} \, \eta(\tau) for \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2,\mathbb{Z}), yielding finite nonzero limits after scaling by the appropriate cusp width. These special values at quadratic irrationals like i and \rho play a crucial role in class number formulas for imaginary quadratic fields \mathbb{Q}(\sqrt{-d}), where the j-invariant j(\tau) for \tau a root of unity or purely imaginary relates algebraic integers to the class number h(-d) through the relation \Delta(\tau) = (2\pi)^{12} \eta(\tau)^{24}, with j(\tau) = 1728 [E_4^3](/page/1728) / \Delta.

Ramanujan-Sato Series Connections

The Dedekind eta function plays a pivotal role in Ramanujan's derivations of rapidly convergent series for $1/\pi, primarily through its special values at quadratic irrationals, which yield class invariants that facilitate these approximations. In his notebooks, Ramanujan expressed such values using eta quotients, linking them to modular equations and elliptic integrals to construct hypergeometric series with exceptional convergence rates. For instance, the value \eta(i)^8 appears in early entries leading to series like \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{n=0}^\infty \frac{(4n)! (1103 + 26390n)}{n!^4 396^{4n}}, which provides about 8 decimal digits of \pi per term. These connections were first systematically explored in Ramanujan's second notebook (page 355) and lost notebook (page 370), with modern proofs relying on eta's transformation properties under the modular group.^[20] Ramanujan's class invariants g_n are defined as g_n = \eta(\tau)^{24/n}, where \tau = \frac{1 + \sqrt{-n}}{2} for positive integers n \equiv 3 \pmod{4}, connecting eta evaluations to singular moduli in imaginary quadratic fields \mathbb{Q}(\sqrt{-n}). These invariants generate the Hilbert class field and satisfy algebraic relations derived from Weber's modular functions, allowing Ramanujan to compute explicit algebraic values like

g_7 = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=4&&&citation_type=wikipedia}}{2} \cdot \frac{\Gamma(1/7)^3 \Gamma(2/7)^3 \Gamma(3/7)^3 \Gamma(4/7)^3}{8\pi^6}^{1/7}

, which he used to parameterize series for $1/\pi. Berndt and colleagues provided rigorous proofs for over 100 such invariants from Ramanujan's notebooks, employing Kronecker's limit formula and eta products to verify their role in cubic, quartic, and higher-degree theories of elliptic functions. The convergence of associated series stems from the eta function's q-expansion, ensuring terms diminish exponentially fast, often yielding dozens of digits per term in optimized forms.^[21] Sato series generalize these Ramanujan formulas, incorporating eta values at quadratic \tau into double sums or hypergeometric expressions for $1/\pi, often tied to higher-level modular forms. Berndt, Chan, and others extended these to arbitrary levels using eta-differentials and McKay-Thompson series, proving convergence via modular invariance and providing explicit multipliers from eta quotients at singular points. This framework has led to ongoing discoveries, such as level-17 and level-20 series, enhancing computational efficiency for \pi.^[20]^[22]^[23]

Eta Quotients

Construction and Modularity Criteria

An eta-quotient of level N is defined as

f(\tau) = \prod_{\delta \mid N} \eta(\delta \tau)^{r_\delta},

where the r_\delta are integers and \eta denotes the Dedekind eta function.^[24] The associated weight is

k = \frac{1}{2} \sum_{\delta \mid N} r_\delta.

^[24] These functions generalize the eta function itself and provide a systematic way to construct modular forms, leveraging the modular properties of \eta. Newman's theorem establishes necessary and sufficient conditions under which an eta-quotient f of level N and weight k is a weakly holomorphic modular form for the principal congruence subgroup \Gamma_1(N).^[25] These conditions are:

\sum_{\delta \mid N} \delta r_\delta \equiv 0 \pmod{24},

\sum_{\delta \mid N} \delta r_{N/\delta} \equiv 0 \pmod{24},

along with a character condition ensuring compatibility of the multiplier system.^[26] Specifically, the associated Dirichlet character is given by \chi(d) = (-1)^k (s / d), where s = \prod_{\delta \mid N} \delta^{r_\delta} and ( \cdot / d ) is the extended Jacobi symbol; \chi must be a genuine Dirichlet character modulo N.^[26] The first congruence arises from the transformation behavior under translations, while the second ensures consistency under the action of the subgroup generators. The character condition guarantees that the overall multiplier system aligns with that of a modular form for \Gamma_1(N). A classic example is the Ramanujan cusp form \Delta(\tau) = \eta(\tau)^{24}, an eta-quotient of level 1 and weight 12 that satisfies the conditions, as \sum \delta r_\delta = 24 \equiv 0 \pmod{24} and the character is trivial.^[25] For half-integer weight, consider \eta(2\tau)^5 \eta(\tau)^{-2} \eta(4\tau)^{-2}, an eta-quotient of level 4 and weight $1/2 proportional to the Jacobi theta function \theta_3(\tau); here, both sums equal 0 modulo 24, and the character matches the required multiplier system for modularity.^[26] The proof of Newman's theorem exploits the multiplicative structure of the eta function's transformation law under \mathrm{SL}(2, \mathbb{Z}): if \eta \mid \gamma = \varepsilon(\gamma) \eta for \gamma \in \mathrm{SL}(2, \mathbb{Z}) and multiplier \varepsilon: \mathrm{SL}(2, \mathbb{Z}) \to \mu_{24}, then for an eta-quotient, f \mid \gamma = \left( \prod \varepsilon(\gamma)^{r_\delta} \right) f, allowing the conditions to ensure this product equals the appropriate character times \chi(\det \gamma)^k.^[25] Newman's original argument relies on detailed computations with the 24th roots of unity in \varepsilon. An elementary proof, bypassing Dedekind sums and using the generation of \Gamma_1(N) by specific matrices, was given by Savitt in 2025.^[26]

Applications in Modular Forms

Eta quotients play a fundamental role in constructing bases for spaces of modular forms M_k(\Gamma_0(N)) for specific levels N, particularly the 121 positive integers N \leq 500 where the graded ring is generated by eta-quotients, as classified by Rouse and Webb (2015).^[3] Newman's theorem provides necessary and sufficient conditions for an eta quotient f(z) = \prod_{0 < \delta \mid N} \eta(\delta z)^{r_\delta} to be a modular form of weight k = \frac{1}{2} \sum r_\delta on \Gamma_1(N), requiring \sum \delta r_\delta \equiv 0 \pmod{24} and \sum (N/\delta) r_\delta \equiv 0 \pmod{24}, along with a compatible nebentypus character.^[26] For N coprime to 6, these conditions simplify due to the property that squares modulo 24 are 1 for integers coprime to 24, ensuring the eta quotient transforms correctly under \Gamma_0(N).^[26] At such levels without elliptic fixed points and with composite structure, eta quotients form a basis for M_k(\Gamma_0(N)) with even integer weight k \geq 2, as the dimension of the space matches the number of such quotients satisfying the conditions.^[3] Hecke operators act on eta quotients by mapping them to scalar multiples of other eta quotients, preserving modularity and often yielding eigenforms. For instance, the eta quotient \eta(z)^{24} = \Delta(z) is a Hecke eigenform under the Hecke operator T_l with eigenvalue 1. More generally, double coset operators T_l transform an eta quotient with multiplier system v to another with compatible system v', enabling the computation of eigenvalues and facilitating the study of Hecke eigenbasis in spaces generated by eta quotients. This action is particularly useful for levels where eta quotients span the space, allowing explicit determination of Hecke eigenvalues through the transformation properties. Eta products appear in decompositions of modular forms involving Eisenstein series, particularly in expressing newforms as linear combinations. For example, in weights 2 and levels such as 30, 33, 35, 38, 40, 42, 44, and 45, every newform can be written as a sum of eta quotients and Eisenstein series of the same level and weight.^[27] Such decompositions leverage the fact that certain eta quotients serve as cusp forms, complementing the non-cuspidal Eisenstein components to span the full space M_k(\Gamma_0(N)). This interplay is essential for computational aspects, as eta quotients provide explicit generators while Eisenstein series contribute the constant terms at infinity.^[27] Borcherds products generalize eta quotients as infinite products over the upper half-plane, playing a central role in monstrous moonshine by constructing the moonshine module for the Monster group. These products, defined using coefficients from weakly holomorphic modular forms, yield denominator identities that match graded dimensions of the Monster's representations, with eta quotients appearing as finite approximations for specific conjugacy classes corresponding to pure A-type Niemeier lattices. In this context, eta quotients define McKay-Thompson series, which are Hauptmoduln for genus zero subgroups, and Borcherds products extend these to capture the full moonshine phenomena through their q-expansions. Despite their utility, eta quotients have limitations when \Gamma_0(N) has elliptic fixed points (of order 2 or 3), which can increase the dimension of M_k(\Gamma_0(N)) beyond what eta quotients alone can span; such points exist for many N, including some coprime to 6 like N=5. Specifically, the number of independent eta quotients is bounded by \dim M_k(\Gamma_0(N)) - \epsilon_2(\Gamma_0(N)) - \epsilon_3(\Gamma_0(N)), where \epsilon_2 and \epsilon_3 count the orbits of these elliptic points, necessitating additional generators like powers of the j-invariant for full generation. For such levels, weakly holomorphic extensions or other forms are required to generate the ring of modular forms.

Extensions and Applications

Analogues and Generalizations

One prominent generalization of the Dedekind eta function extends to Hecke groups, which are subgroups of SL(2,ℝ) generated by certain hyperbolic elements beyond the modular group SL(2,ℤ). In a 2025 construction, an analogue η_D(τ) is defined for the Hecke group H(√D), where D > 5 is a fundamental discriminant congruent to 1 modulo 4 corresponding to a real quadratic field, as η_D(τ) = q^m \prod_{n=1}^\infty \left[ (1 - q^n) \chi_D(n) \prod_{a=1}^D (1 - e^{2\pi i a / D} q^n) \chi_D(a) \right] with q = e^{2\pi i \tau / \sqrt{D}}, m = -L(-1, \chi_D)/2, and χ_D the primitive real quadratic character modulo D.^[28] This function exhibits modular properties under the action of H(√D), yielding a family of holomorphic modular functions analogous to the classical eta's transformation behavior, and it connects to partition theory via quadratic characters.^[28] Generalizations to metaplectic groups involve lifting the eta function to half-integral weights on the double cover of SL(2,ℝ), known as the metaplectic group Mp_2(ℝ). These extensions, developed in Shimura's framework, allow the eta function to serve as a building block for modular forms of weight 1/2, incorporating a cocycle factor to account for the double covering. Such forms transform under the metaplectic representation, enabling applications to vector-valued modular forms and theta series with half-integral weights. For real quadratic fields, analogues of the eta function arise in the context of indefinite binary quadratic forms, contrasting the definite forms underlying the classical case. These constructions adapt the eta product to indefinite theta series associated with units in the field, leading to meromorphic functions with transformation laws under the corresponding Hecke groups or Atkin-Lehner involutions.^[29] Key examples include limit formulas linking these eta analogues to Dedekind zeta functions of real quadratic fields, providing analytic continuations and functional equations for indefinite settings.^[30] Imaginary powers of the Dedekind eta function, denoted η(τ)^{i t} for real t, represent another generalization, with their Fourier coefficients exhibiting zeros that follow specific recursive patterns. A 2018 study analyzes the distribution of these zeros on the imaginary axis, showing that they label polynomials whose roots correspond to values of t where the nth Fourier coefficient vanishes, linking to Hurwitz-stable polynomials and analytic number theory.^[31] This approach reveals asymptotic behaviors and connections to the eta function's q-expansion, without altering its modular weight.^[31] New approaches to level 6 identities for the Dedekind eta function build on Somos's conjectures, providing proofs via theta function relations and modular equations. In 2021 work, two such identities are established using alternative methods, including quadratic transformations and eta quotients of level 6, which refine Somos's computational discoveries and extend to higher-level generalizations without introducing new groups.^[13] These proofs highlight bilinear relations among eta products, offering insights into the algebraic structure of level 6 modular forms.^[13]

Uses in Physics and Recent Advances

In theoretical physics, the Dedekind eta function plays a prominent role in the partition function of bosonic string theory. Specifically, the partition function for the 24 transverse bosonic degrees of freedom on the worldsheet torus is given by Z = \left( \frac{1}{\eta(\tau)} \right)^{24}, where \tau is the complex structure modulus of the torus.^[32] This expression arises from the infinite product representation of the eta function, which regularizes the sum over oscillator modes, ensuring modular invariance under SL(2,ℤ) transformations.^[33] The modular discriminant \Delta(\tau) = (2\pi)^{12} \eta(\tau)^{24} further connects this to the cusp form of weight 12, encapsulating the anomaly cancellation in critical dimension 26.^[32] The eta function also appears in the representation theory of affine Lie algebras, particularly in the characters of integrable highest-weight modules for Kac-Moody algebras. The Weyl-Kac character formula expresses these characters as ratios involving the theta function of the root lattice and a denominator featuring powers of the eta function, such as \eta(\tau)^r where r is the rank of the algebra.^[34] For affine algebras like \mathfrak{sl}(2,\mathbb{C}) at level k, the string functions—graded dimensions of modules—are quotients of eta products, yielding identities like the Rogers-Ramanujan generalizations.^[35] These structures underpin conformal field theories associated with affine symmetries, linking algebraic representations to modular forms.^[36] In the context of monstrous moonshine, eta quotients feature in the Fourier expansions of modular functions related to the Monster simple group. The j-function, central to moonshine, admits an expansion involving $1 / \Delta(\tau), where \Delta(\tau) = \eta(\tau)^{24}, and more generally, eta products parameterize the Hauptmoduln for genus-zero congruence subgroups appearing in moonshine modules.^[37] Borcherds' proof of the moonshine conjectures relies on vertex operator algebras whose characters are eta quotients, establishing graded traces over Monster representations that match the j-function coefficients.^[38] This connection highlights eta's role in bridging finite group theory with infinite-dimensional Lie structures. Recent mathematical advances have focused on automorphy properties of the eta function through generalized divisor sums. In a 2025 study, the case \alpha = 1 in the family of functions \sum_{n=1}^\infty \sigma_\alpha(n) q^n, where \sigma_1(n) is the sum-of-divisors function, provides a pathway to proving the automorphy of \eta(\tau) in the spirit of Ramanujan's classical assertions, extending beyond the \alpha = 0 divisor case.^[39] Additionally, evaluations of eta products have been explored in the context of two-dimensional zeta functions, such as Epstein zeta functions associated with quadratic forms, yielding closed forms for lattice sums via eta identities at rational arguments. Ongoing challenges include extensions of eta quotient theorems to levels N not coprime to 6, where classical criteria like Newman's theorem require adjustments for the presence of 2 and 3 in the conductor.^[40] Clarifications on the role of Dirichlet characters \chi(n) in these cases remain incomplete, particularly for non-principal characters modulo N sharing factors with 6, hindering full modularity classifications.^[41]

References

[1]
https://arxiv.org/pdf/2302.03280.pdf
[2]
Schreiben an Herrn Borchardt über die Theorie der elliptischen ...
Dec 14, 2009 · Schreiben an Herrn Borchardt über die Theorie der elliptischen Modul-Functionen. R. Dedekind. R. Dedekind. Search for this author in: De ...
[3]
[PDF] Spaces of modular forms spanned by eta-quotients
In 1877, Richard Dedekind [3] defined the function. (1) η(z) = q1/24. ∞. Y n=1 ... Dedekind eta-function. The reciprocal of the Dedekind eta-function is.
[4]
[PDF] The Pentagonal Number Theorem and Modular Forms
Mar 28, 2010 · So at last the calculation of the length of an ellipse appears imminent. If we solve the equation of an ellipse for y, we get y = b q. 1 − x2.
[5]
[PDF] MODULAR FORMS LECTURE 16: THE DEDEKIND ETA FUNCTION
Then we would have the Dedekind eta function: η(τ) := q. 1 24 (q)∞. If this were modular, we'd expect the weight to be 1/2. But, no modular forms of weight.Missing: mathematics | Show results with:mathematics
[6]
Modular Discriminant -- from Wolfram MathWorld
The modular discriminant is defined by Delta(tau)=(2pi)^(12)qproduct_(r=1)^infty(1-q^r)^(24). However, some care is needed as some authors omit the factor of ( ...Missing: original source
[7]
Dedekind Eta Function -- from Wolfram MathWorld
The Dedekind eta function is defined over the upper half-plane H={tau:I[tau]>0} by (OEIS A010815), where q^_=e^(2piitau) is the square of the nome q.
[8]
An Elementary Proof of the Transformation Formula for the Dedekind ...
Feb 7, 2023 · Abstract page for arXiv paper 2302.03280: An Elementary Proof of the Transformation Formula for the Dedekind Eta Function.Missing: laws SL(
[9]
A Course in Arithmetic - SpringerLink
This book is divided into two parts. The first one is purely algebraic. Its objective is the classification of quadratic forms over the field of rational ...Missing: URL | Show results with:URL
[10]
https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-174/issue-1/Modular-invariance-and-uniqueness-of-conformal-characters/cmp/1104275096.pdf
[11]
Mathematical Physics - Project Euclid
. Finally, recall that the Dedekind eta function η is a modular form of weight 1/2 for DΓ, more precisely, that there exists a one- dimensional ...
[12]
[math/0510054] Euler and the pentagonal number theorem - arXiv
Oct 3, 2005 · In this paper we give the history of Leonhard Euler's work on the pentagonal number theorem, and his applications of the pentagonal number theorem.
[13]
[PDF] 8. PARTITIONS George E. Andrews
This formula is now known as Euler's Pentagonal Number Theorem. We shall examine Fabian Franklin's proof of it [31] in the next section.
[14]
New approach to Somos's Dedekind eta-function identities of level 6
In the present work, we prove few new Dedekind eta-function identities of level 6 discovered by Somos in two different methods.
[15]
Ramanujan's identities for eta-functions | Mathematische Annalen
Ramanujan's identities for eta-functions ... Biagioli, A.J.F.: The construction of modular forms as products of transforms of the Dedekind eta function.
[16]
Jacobi Triple Product -- from Wolfram MathWorld
The Jacobi triple product is the beautiful identity product_(n=1)^infty(1-x^(2n))(1+x^(2n-1)z^2)(1+(x^(2n-1))/(z^2))=sum_(m=-infty)^inftyx^(m^2)z^(2m).Missing: eta | Show results with:eta
[17]
[PDF] eta-quotients and theta functions - Lemke Oliver
... Dedekind's eta-function η(z), defined via an infinite product, giving it as a certain kind of infinite sum known as a theta function. Using the theory of.
[18]
[PDF] arXiv:2302.03280v1 [math.NT] 7 Feb 2023
Feb 7, 2023 · The Dedekind eta function is introduced by Dedekind in 1877 and is defined in the upper half plane H = {τ | Imτ > 0} by the equation η(τ) ...Missing: original | Show results with:original
[19]
Carl Jacobi (1804 - 1851) - Biography - MacTutor
In 1834 Jacobi proved that if a single-valued function of one variable is doubly periodic then the ratio of the periods is non-real. This result prompted much ...Missing: eta Dedekind
[20]
https://mrc.sdu.edu.cn/ziliao/58.pdf
[21]
[PDF] POWERS OF THE DEDEKIND ETA FUNCTION AND HURWITZ ...
... Dedekind eta function \eta, studied first by Dedekind. This makes it possible to apply the theory of modular forms, which includes the Hecke theory. Let ...Missing: original | Show results with:original
[22]
[PDF] Ramanujan's Series for 1/π: A Survey
Mar 31, 2009 · 7. N. D. Baruah and B. C. Berndt,, Ramanujan's series for 1/π arising from his cubic and quartic theories of elliptic functions, J. Math.
[23]
[PDF] Ramanujan's class invariants, Kronecker's Limit Formula,and ...
Abstract. In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated.Missing: g_n | Show results with:g_n
[24]
[PDF] Ramanujan-type formulae for 1/π: A second wind? - arXiv
May 14, 2008 · is the Dedekind eta-function. The function g(τ) = (2πi)−1f′(τ)/f(τ) ... Liu, 'Domb's numbers and. Ramanujan–Sato type series for 1/π', Adv.
[25]
Series for 1/π of level 20 - ScienceDirect.com
Properties of theta functions and Eisenstein series dating to Jacobi and Ramanujan are used to deduce differential equations associated with McKay Thompson ...
[26]
https://arxiv.org/abs/2507.16225
[27]
https://arxiv.org/abs/1604.07774
[28]
[2507.16225] An elementary proof of Newman's eta-quotient theorem
Jul 22, 2025 · Newman studied the modularity of eta-quotients, giving necessary and sufficient conditions for a function of the form \prod_{0 < m | N} eta(mz)^ ...
[29]
[1604.07774] Eta quotients, Eisenstein series and Elliptic Curves
Apr 26, 2016 · We express all the newforms of weight 2 and levels 30, 33, 35, 38, 40, 42, 44, 45 as linear combinations of eta quotients and Eisenstein series.Missing: decompositions | Show results with:decompositions
[30]
An Analogue of the Dedekind Eta Function for Hecke Groups ... - arXiv
We construct an analogue of the classical Dedekind eta function for the Hecke group H(\sqrt{D}). This gives rise to a family of holomorphic ...
[31]
[PDF] A Kronecker limit formula for real quadratic fields
real quadratic field K (a Dirichlet series involving an indefinite quadratic form) ... Dedekind's eta-function. Proof. The terms with m = 0 clearly give. ~,. 1 ...
[32]
[PDF] Indefinite Theta Functions and Zeta Functions - Gene S. Kopp
Indefinite Zeta Functions and Real Quadratic Fields . ... Dedekind eta function. Here Log fp is the branch satisfying. (1.33). (Log fp)(τ) ∼ πi p2. 1 − p1 +.
[33]
Imaginary Powers of the Dedekind Eta Function
Jun 4, 2018 · 1. The main purpose of this article is to study the distribution of the roots of a family of recursively defined polynomials pn(x).Missing: zeros | Show results with:zeros
[34]
[PDF] String Theory - DAMTP
The function appearing in the partition function for the scalar field has a name: it is the inverse of the Dedekind eta function η(q) = q1/24. ∞. Y n=1. (1 ...<|separator|>
[35]
[PDF] Finite Temperature Strings - arXiv
We stress the relation between the level density p(N) of the bosonic string, the Dedekind eta function η(τ) and Ramanujan's τ-function τ(n).
[36]
Hall-Littlewood polynomials and characters of affine Lie algebras
The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is ...
[37]
On string functions and double-sum formulas - ADS
Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions.<|separator|>
[38]
Hall–Littlewood polynomials and characters of affine Lie algebras
The Weyl–Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac–Moody algebras.
[39]
[PDF] Monstrous Moonshine: The first twenty-five years - arXiv
Apr 14, 2004 · In §2.2 we focus on the modular groups and functions which arise in Monstrous Moonshine. 2.1. The Monster. By definition, a simple group is one ...
[40]
Mathematical Physics Monstrous Moonshine from Orbifolds
We show that the main Conway and Norton Monstrous Moonshine properties, stating that the. Thompson series for each Monster group conjugacy class has a modular.<|separator|>
[41]
Mathematics, Volume 13, Issue 11 (June-1 2025) – 206 articles
The α=0 case is the divisor function, while the α=1 case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan. Full article. (This ...
[42]
[PDF] an elementary proof of newman's eta-quotient theorem
Abstract. Let η(z) be the Dedekind eta function. Newman [New1, New2] studied the modularity of eta-quotients, giving necessary and sufficient condi-.<|control11|><|separator|>
[43]
New proofs of some Dedekind η-function identities of level 6
Sep 12, 2025 · Here, in this article, we make use of the modular equation of degree 3 in two methods. We further give some interesting combinatorial ...Missing: automorphy | Show results with:automorphy