Dedekind Eta Function Research Articles

Lattice sums of I-Bessel functions, theta functions, linear codes and heat equations

We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by I-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of I-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that I-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{Z}}^n$$\\end{document} whose initial condition is given by a linear code.

Minimal type-I Dirac seesaw and leptogenesis under A4 modular invariance

We present a Dirac mass model based on A4 modular symmetry within Type-I seesaw framework. This extension of Standard Model requires three right-handed neutrinos and three heavy Dirac fermions superfields, all singlet under SU(2)L symmetry. The scalar sector is extended by the inclusion of a SU(2)L singlet superfield, χ. Here, the modular symmetry plays a crucial role as the Yukawa couplings acquire modular forms, which are expressed in terms of Dedekind eta function η(τ). Therefore, the Yukawa couplings follow transformations akin to other matter fields, thereby obviating the necessity of additional flavon fields. The acquisition of vev by complex modulus τ leads to the breaking of A4 modular symmetry. We have obtained predictions on neutrino oscillation parameters, for example, the normal hierarchy for the neutrino mass spectrum. Furthermore, we find that heavy Dirac fermions, in our model, can decay to produce observed baryon asymmetry of the Universe through Dirac leptogenesis.

Coupled free fermion conformal field theories

We study a specific class of CFTs that involve coupled free fermions, arising from parafermion CFTs and lattice constructions. We analyse their representation spaces and the underlying exclusion statistics of coupled free fermions using specific bases. In one particular case, we reveal an unexpected connection between the coset construction of parafermions, the lattice construction, and the orbifold thereof. This connection is supported by proving a range of character identities within the context of coupled free fermions. Simultaneously, we obtain explicit expressions of certain string functions in terms of Dedekind eta functions.

On the Derivatives of Eta Quotients of Level Eighteen

On the Derivatives of Eta Quotients of Level Eighteen

Theta blocks

We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta function and show that they give a new powerful method to construct Jacobi forms and Siegel modular forms, with applications also in lattice theory and algebraic geometry. One of the central questions is when a theta block defines a Jacobi form. It turns out that this seemingly simple question is connected to various deep problems in different fields ranging from Fourier analysis over infinite-dimensional Lie algebras to the theory of moduli spaces in algebraic geometry. We give several answers to this question.

The Generalized Eta Transformation Formulas as the Hecke Modular Relation

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke’s work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions.

An Elementary Proof of the Transformation Formula for the Dedekind Eta Function

In this work, we give an elementary proof of the transformation formula for the Dedekind eta function under the action of the modular group $\text{PSL}\,(2,\mathbb{Z})$. We start by giving a proof of the transformation formula $\eta(\tau)$ under the transformation $\tau\to -1/\tau$, using the Jacobi triple product identity and the Poisson summation formula. After we establish some identities for the Dedekind sum, the transformation formula for $\eta(\tau)$ under the transformation induced by a general element of the modular group $\text{PSL}\,(2,\mathbb{Z})$ is derived by induction.

Zeros transfer for recursively defined polynomials

The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.

On the vanishing of the coefficients of CM eta quotients

AbstractThis work characterizes the vanishing of the Fourier coefficients of all CM (Complex Multiplication) eta quotients. As consequences, we recover Serre’s characterization about that of $\eta(12z)^{2}$ and recent results of Chang on the pth coefficients of $\eta(4z)^{6}$ and $\eta(6z)^{4}$. Moreover, we generalize the results on the cases of weight 1 to the setting of binary quadratic forms.

Kronecker's first limit formula for Kleinian groups

We prove Kronecker's first limit formula for cofinite discrete subgroups of SL$(2,\mathbbC)$, also called Kleinian groups, generalizing a method of Goldstein over SL$(2,\mathbb R)$. The proof uses the Fourier expansion of Eisenstein series, leading to an analogue of the logarithm of the Dedekind eta function, and whose transformation law produces a function analogous to the one related to Dedekind sums. We derive several basic properties of this new function.

Bounds for orders of zeros of a class of Eisenstein series and their applications on dual pairs of eta quotients

Let k k be an even positive integer, p p be a prime and m m be a nonnegative integer. We find an upper bound for orders of zeros (at cusps) of a linear combination of classical Eisenstein series of weight k k and level p m p^m . As an immediate consequence we find the set of all eta quotients that are linear combinations of these Eisenstein series and hence the set of all eta quotients of level p m p^m whose derivatives are also eta quotients.

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

AbstractFor a positive integer N, let be the modular curve over and its Jacobian variety. We prove that the rational cuspidal subgroup of is equal to the rational cuspidal divisor class group of when for any prime p and any squarefree integer M. To achieve this, we show that all modular units on can be written as products of certain functions , which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on under a mild assumption.

On string functions and double-sum formulas

String functions are important building blocks of characters of integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions. We produce new relations between string functions by writing them as double-sums and then using certain symmetry relations. We evaluate the series using special double-sum formulas that express Hecke-type double-sums in terms of Appell--Lerch functions and theta functions, where we point out that Appell--Lerch functions are the building blocks of Ramanujan's classical mock theta functions.

Type III seesaw under A_4 modular symmetry with leptogenesis

We make an attempt to study neutrino phenomenology in the framework of type-III seesaw by considering A_4 modular symmetry in the super-symmetric context. In addition, we have included a local U(1)_{B-L} symmetry which eventually helps us to avoid certain unwanted terms in the superpotential. Hitherto, the seesaw being type-III, it involves three fermion triplet superfields Sigma _R, along with which, we have included a singlet weighton field (rho ). In here, modular symmetry plays a crucial role by avoiding the usage of excess flavon (weighton) fields. Also, the Yukawa couplings acquire modular forms which are expressed in terms of Dedekind eta function eta (tau ). However, for numerical analysis we use q expansion expressions of these couplings. Therefore, the model discussed here is triumphant enough to accommodate the observed neutrino oscillation data and also successfully explains observed baryon asymmetry of the universe through leptogenesis.

The Efficacy of Therapeutic Use of Play on Improving Attention Span of Mild Intellectually Disabled Children

Objectives: This study aimed to determine the efficacy of the therapeutic use of play on improving the attention span of children with Mild Intellectual Disability (MID) in elementary school. Methods: This was a quasi-experimental study with a pre-test-post-test with a control group so that 30 girls with MID were selected from an exceptional school in Tehran City, Iran, via convenience sampling method and assigned to the experimental and control groups. The attention span of all subjects was assessed by the Toulouse-Pieron test and the Cognitive Diagnostic Battery (CDB) before and after the therapeutic sessions. The experimental group participated in 16 therapeutic play sessions (35 minutes for each session) for 8 weeks, but the control group did not. The analysis of covariance (ANCOVA) was used to analyze the data. Results: The mean±SD age were 9.80±1.32 and 9.73±1.22 for the experimental and control groups, respectively. In addition, the mean attention span in the experimental group varied from 7.60 to 18.78 after the therapeutic sessions. In addition, after the therapeutic sessions, a significant difference (P<0.001) was observed between the experimental and control groups relating to attention span. According to the Eta quotient, 94% of the variation in the attention span of the experimental group is due to participating in the intervention sessions. Discussion: Therapeutic use of play is an effective method to improve the attention span of children with MID. Therefore, the therapeutic use of play can be applied to improve the academic performance of students with MID and reduce negative outcomes due to inattention.

Congruences for Fourier coefficients of eta‐quotients modulo powers of 5, 7, 11, 13, and 17

In this paper, we investigate the Fourier coefficients of the eta‐quotients of the forms where is the Dedekind eta function, , and 17; is a positive integer, and are arbitrary integers. We prove Ramanujan's type congruences for the Fourier coefficients of modulo powers of prime . We recover several results due to Atkin, Garvan, Gordon, Wang, Mestrige, and others. We give few examples, and we establish an improvement for Wang's results related to the 11‐regular partition function.

The Effectiveness of Cognitive Rehabilitative Program on Executive Functions in Children With Dyscalculia

Objective Students’ academic success is crucial for families, educators, and the community. Specific learning disorder reduces students’ academic performance and makes education difficult. Because of the rising prevalence of learning disorders (especially dyscalculia), it is necessary to pay more attention to dyscalculia, its diagnosis, and assessment to develop new interventions and scientific strategies to reduce symptoms. Negative consequences of math problems in early childhood probably affect the individual and social aspects of life until adulthood. As far as we know, most children with dyscalculia have significant problems in executive functions, so the application of effective educational methods on executive functions will improve cognitive functions and learning. The present study was conducted to determine the effectiveness of a cognitive rehabilitation program on the executive functions of students with dyscalculia. Materials & Methods This research was a quasi-experimental study with a pre-test-post-test design with a control group. The statistical population consisted of all 8 to 12 years old students who were studying in specific schools for learning disorders in the 2020-2021 academic year in Sanandaj City, Iran. Thirty children (17 boys and 13 girls) were selected by convenience method and assessed using the Iranian Key Math Test. Then, 28 students with one standard deviation lower than the mean (≤85 points) were diagnosed as dyscalculic, and their parents signed a written consent form. Students were excluded if they had seizures and epilepsy, or other neurodevelopmental disorders or were absent more than two times from intervention sessions. All students were matched according to gender and age and randomly assigned to the experimental (7 boys and 7 girls) and control groups (8 boys and 6 girls). The executive functions of all students were assessed by the Behavioral Rating Inventory for Executive Functions (BRIEF). The experimental group individually participated in 12 sessions (twice a week; 30 to 45 minutes per session) and was trained by the cognitive rehabilitation program. However, the control group only attended the mainstream curriculum. Again, all students were assessed by the BRIEF in the last session and 5 weeks later in the follow-up. The obtained data were analyzed by using analysis of variance with repeated measures. Results The findings showed that the cognitive rehabilitation program has significantly influenced executive functions and their components (behavior regulation and metacognition) in students with dyscalculia. According to the eta quotient, 42%, 32%, and 43% of the variances of executive functions, behavior regulation, and metacognition of the experimental group, respectively, have been explained by participating in the cognitive rehabilitation program. In addition, the program’s effectiveness lasted for a 5-week follow-up later. Conclusion Considering the effectiveness of cognitive rehabilitation programs on executive functions, designing and implementing the same rehabilitative programs not only improves the executive functions of students with dyscalculia but also prevents their problems relating to learning academic skills.

The smallest parts function associated with ω(q)

We establish two families of congruences modulo powers of 5 for the Fourier coefficients of [Formula: see text], where [Formula: see text] is the weight 2 Eisenstein series and [Formula: see text] is the Dedekind eta function. This allows us to prove similar congruences for two smallest parts functions. The first function is [Formula: see text], which was introduced by Andrews, Dixit and Yee and is associated with Ramanujan/Watson’s third-order mock theta function [Formula: see text]. The second one is [Formula: see text], which appeared in the work of Garvan and Jennings-Shaffer. Moreover, we confirm two conjectural congruences of Wang.

Moments of ranks and cranks, and quotients of Eisenstein series and the Dedekind eta function

Atkin and Garvan introduced the functions Nk(n) and Mk(n), which denote the k-th moments of ranks and cranks in the theory of partitions. Let e2r(n) be the n-th Fourier coefficient of E2r(τ)/η(τ), where E2r(τ) is the classical Eisenstein series of weight 2r and η(τ) is the Dedekind eta function. Via the theory of quasi-modular forms, we find that for k≤5, Nk(n) and Mk(n) can be expressed using e2r(n) (0≤r≤k), p(n) and N2(n). For k>5, additional functions are required for such expressions. For r∈{2,3,4,5,7}, by studying the action of Hecke operators on E2r(τ)/η(τ), we provide explicit congruences modulo arbitrary powers of primes for e2r(n). Moreover, for ℓ∈{5,7,11,13} and any k≥1, we present uniform methods for finding nice representations for ∑n=0∞e2r(ℓkn+124)qn, which work for every r≥2. These representations allow us to prove congruences modulo powers of ℓ, and we have done so for e4(n) and e6(n) as examples. Based on the congruences satisfied by e2r(n), we establish congruences modulo arbitrary powers of ℓ for the moments and symmetrized moments of ranks and cranks as well as higher order spt-functions.

Explicit transformations of certain Lambert series

An exact transformation, which we call the master identity, is obtained for the first time for the series \(\sum _{n=1}^{\infty }\sigma _{a}(n)e^{-ny}\) for \(a\in {\mathbb {C}}\) and Re\((y)>0\). New modular-type transformations when a is a nonzero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan’s famous formula for \(\zeta (2m+1)\). The Wigert–Bellman identity arising from the \(a=0\) case of the master identity is derived too. When a is an odd integer, the well-known modular transformations of the Eisenstein series on \(SL _{2}\left( {\mathbb {Z}}\right) \), that of the Dedekind eta function as well as Ramanujan’s formula for \(\zeta (2m+1)\) are derived from the master identity. The latter identity itself is derived using Guinand’s version of the Voronoï summation formula and an integral evaluation of N. S. Koshliakov involving a generalization of the modified Bessel function \(K_{\nu }(z)\). Koshliakov’s integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new modular-type transformation involving the sums-of-squares function \(r_k(n)\). Some results on functions self-reciprocal in the Watson kernel are also obtained.