Dedekind Eta Function Research Articles

Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series

Let E(z, s) be the non-holomorphic Eisenstein series for the modular group $${\mathrm{SL}}(2,{{\mathbb {Z}}})$$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $$s=1$$ of E(z, s) is essentially the logarithm of the Dedekind eta function. This eta function is a weight 1/2 modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $${\Gamma }$$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map $$S:{\Gamma }\rightarrow {{\mathbb {R}}}$$ which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from $${\Gamma }$$ to $${{\mathbb {C}}}$$ produces higher-order Eisenstein series. These series share many of the properties of E(z, s) though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher-order Dedekind symbols $$S^*:{\Gamma }\rightarrow {{\mathbb {R}}}$$. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups $${\Gamma }_0(N)$$ are rational.

On the computation of identities relating partition numbers in arithmetic progressions with eta quotients: An implementation of Radu's algorithm

In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg, in which the generating functions of a partition function over a given set of arithmetic progressions are expressed in terms of Dedekind eta quotients over a given congruence subgroup. These identities include the famous results by Ramanujan which provide a witness to the divisibility properties of p(5n+4), p(7n+5). We give an implementation of this algorithm using Mathematica. The basic theory is first described, and an outline of the algorithm is briefly given, in order to describe the functionality and utility of our package. We thereafter give multiple examples of applications to recent work in partition theory. In many cases we have used our package to derive alternate proofs of various identities or congruences; in other cases we have improved previously established identities.

A Generalization of the Secant Zeta Function as a Lambert Series

Recently, Lalín, Rodrigue, and Rogers have studied the secant zeta function and its convergence. They found many interesting values of the secant zeta function at some particular quadratic irrational numbers. They also gave modular transformation properties of the secant zeta function. In this paper, we generalized secant zeta function as a Lambert series and proved a result for the Lambert series, from which the main result of Lalín et al. follows as a corollary, using the theory of generalized Dedekind eta-function, developed by Lewittes, Berndt, and Arakawa.

Arithmetic properties derived from coefficients of certain eta quotients

For a positive integer k, let F(q)k:=∏n≥1(1−qn)4k(1+q2n)2k=∑n≥0ak(n)qn\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ F (q)^{k}:= \\prod_{n \\geq 1} \\frac{(1-q^{n})^{4k}}{(1+q^{2n})^{2k}} = \\sum_{n\\geq 0} \\frak{a}_{k} (n)q^{n} $$\\end{document} be the eta quotients. The coefficients frak{a}_{1} (n) can be interpreted as a certain kind of restricted divisor sums. In this paper, we give the signs and modulo values for frak{a}_{1} (n) and frak{a}_{2} (m) and calculate several convolution sums involving frak{a}_{k} (n).

Zeros of recursively defined polynomials

Let g, h be two arithmetic functions. In this paper, we study zero-free intervals of polynomials of the type We use a difference equation approach. Let be the divisor sum function and the identity function, then the zeros of the polynomials dictate the vanishing properties of the coefficients of all powers of the Dedekind eta function. This is directly related to the Lehmer conjecture on the non-vanishing of the Ramanujan τ-function. Let be the identity function. Then, the polynomials are related to generalized Laguerre polynomials for and to Chebyshev polynomials of the second kind for . The paper is furnished with several interesting examples. We found polynomials with decreasing minimal real zeros. We provide a simple arithmetic function g to approximate the minimal real zeros attached to σ.

Extremal determinants of Laplace\u2013Beltrami operators for rectangular tori

In this work we study the determinant of the Laplace–Beltrami operator on rectangular tori of unit area. We will see that the square torus gives the extremal determinant within this class of tori. The result is established by studying properties of the Dedekind eta function for special arguments. Refined logarithmic convexity and concavity results of the classical Jacobi theta functions of one real variable are deeply involved.

ON A LATTICE GENERALISATION OF THE LOGARITHM AND A DEFORMATION OF THE DEDEKIND ETA FUNCTION

We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty )$, initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms.

The Dedekind eta function and D’Arcais-type polynomials

D’Arcais-type polynomials encode growth and non-vanishing properties of the coefficients of powers of the Dedekind eta function. They also include associated Laguerre polynomials. We prove growth conditions and apply them to the representation theory of complex simple Lie algebras and to the theory of partitions, in the direction of the Nekrasov–Okounkov hook length formula. We generalize and extend results of Kostant and Han.

The generalized modified Bessel function and its connection with Voigt line profile and Humbert functions

Recently Dixit, Kesarwani, and Moll introduced a generalization Kz,w(x) of the modified Bessel function Kz(x) and showed that it satisfies an elegant theory similar to that of Kz(x). In this paper, we show that while K12(x) is an elementary function, K12,w(x) can be written in the form of an infinite series of Humbert functions. As an application of this result, we generalize the transformation formula for the logarithm of the Dedekind eta function η(z). We also establish a connection between K12,w(x) and the cumulative distribution function corresponding to the Voigt line profile.

Neutrino mass and mixing with A5 modular symmetry

We present a comprehensive analysis of neutrino mass and lepton mixing in theories with $A_5$ modular symmetry. We construct the weight 2, weight 4 and weight 6 modular forms of level 5 in terms of Dedekind eta-functions and Klein forms, and their decomposition into irreducible representation of $A_5$. We construct all the simplest models based on $A_5$ modular symmetry, including scenarios of models with and without flavons in the charged lepton sectors. For each case, the neutrino masses can be generated through either the Weinberg operator or the type I seesaw mechanism. We perform an exhaustive numerical analysis, organising our results in an extensive set of figures and tables.

Neutrino masses and mixing from double covering of finite modular groups

We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level N can be arranged into irreducible representations of the homogeneous finite modular group {Gamma}_N^{prime } which is the double covering of ΓN. The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of {Gamma}_3^{prime } ≅ T′. The modular forms of weights 2, 3, 4, 5 and 6 are presented. We build a model of lepton masses and mixing based on T′ modular symmetry.

When is the derivative of an eta quotient another eta quotient?

In this paper we use techniques from the theory of modular forms to determine all eta quotients whose derivative is also an eta quotient of levels up to 36. In addition, we present an algorithm that determines all eta quotients in M2k(Γ0(N)). We also discuss some applications of these results. In particular, we evaluate a number of integrals in terms of algebraic constants.

Maximal abelian extension of $$X_0(p)$$ unramified outside cusps

Let p be a prime number. Mazur proved that a geometrically maximal unramified abelian covering of \(X_0(p)\) over \(\mathbb {Q}\) is given by the Shimura covering \(X_2(p) \rightarrow X_0(p)\), that is, a unique subcovering of \(X_1(p) \rightarrow X_0(p)\) of degree \(N_p := (p-1)/\gcd (p-1, 12)\). In this short paper, we show that a geometrically maximal abelian covering \(X_2'(p) \rightarrow X_0(p)\) of \(X_0(p)\) over \(\mathbb {Q}\) unramified outside cusps is cyclic of degree \(2N_p\). The main ingredient for the construction of \(X_2'(p)\) is the generalized Dedekind eta functions.

Recurrence relations for polynomials obtained by arithmetic functions

Families of polynomials associated to arithmetic functions [Formula: see text] are studied. The case [Formula: see text], the divisor sum, dictates the non-vanishing of the Fourier coefficients of powers of the Dedekind eta function. The polynomials [Formula: see text] are defined by [Formula: see text]-term recurrence relations. For the case that [Formula: see text] is a polynomial of degree [Formula: see text], we prove that at most a [Formula: see text] term recurrence relation is needed. For the special case [Formula: see text], we obtain explicit formulas and results.

Extension of results of Kostant and Han

Kostant proved that the mth coefficient of the $$(r^2-1)$$ th power of the Dedekind eta function is non-vanishing for $$r \ge \mathsf {max} \{4,m \}$$ and r an integer. This was generalized by Han to the real case applying the Nekrasov–Okounkov hook length formula. We extend these results to the complex case and identify coefficients of the zth power with values of polynomials $$P_m(-z)$$ . We assume that all non-trivial roots have a negative real part. Numerical data indicate that the quadratic bound can be replaced by a linear bound. Han showed by a reversion method that $$P_{m} \left( m +1\right) $$ is divisible by $$m +1$$ . We extend this result to $$P_{m} \left( \pm n \right) $$ , where m and n are two coprime positive integers. We obtain congruences modulo n in the case $$P_p\left( \pm n \right) $$ , where p is a prime. This leads to new non-vanishing results for the pth power of the Dedekind eta function, not covered in the theorems of Kostant and Han. We illustrate our results with numerical data and examples.

On conjectures regarding the Nekrasov–Okounkov hook length formula

The Nekrasov–Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The first conjecture is a refined Nekrasov–Okounkov formula involving hooks with trivial legs. We give a proof of the conjecture. The second conjecture is on properties of the roots of the underlying D’Arcais polynomials. We give a counterexample and present a new conjecture. The third conjecture is on the unimodality of the coefficients of the involved polynomials. We confirm the conjecture up to the polynomial degree 1000.

Generalized Lambert series and arithmetic nature of odd zeta values

AbstractIt is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ.

Eta quotients of level 12 and weight 1

We find all the eta quotients in the spaces $M_1 \Big(\Gamma_0(12), \left(\frac{d}{\cdot}\right) \Big)$ ($d=-3, -4$) of modular forms and determine their Fourier coefficients, where $\left(\frac{d}{\cdot}\right)$ is the Legendre-Jacobi-Kronecker symbol.

The asymptotic formulas for coefficients and algebraicity of Jacobi forms expressed by infinite product

We determine asymptotic formulas for the Fourier coefficients of Jacobi forms expressed by infinite products with Jacobi theta functions and the Dedekind eta function. These are generalizations of results about the growth of the Fourier coefficients of Jacobi forms given by an inverse of Jacobi theta function to derive the asymptotic behavior of the Betti numbers of the Hilbert scheme of points on an algebraic surface by Bringmann–Manschot and about the asymptotic behavior of the χ y -genera of Hilbert schemes of points on K3 surfaces by Manschot–Rolon. We also get the algebraicity of the generating functions given by Göttsche for the Hilbert schemes associated to general algebraic surfaces.

GENERALIZED LAMBERT SERIES, RAABE’S COSINE TRANSFORM AND A GENERALIZATION OF RAMANUJAN’S FORMULA FOR

A comprehensive study of the generalized Lambert series$\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},0<a\leqslant 1,~x>0$,$N\in \mathbb{N}$and$h\in \mathbb{Z}$, is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan’s famous formula for$\unicode[STIX]{x1D701}(2m+1)$for$m>0$, the transformation formula for the logarithm of the Dedekind eta function and Wigert’s formula for$\unicode[STIX]{x1D701}(1/N),N$even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series$E_{2}(z)$and$E_{2}(-1/z)$. An identity relating$\unicode[STIX]{x1D701}(2N+1),\unicode[STIX]{x1D701}(4N+1),\ldots ,\unicode[STIX]{x1D701}(2Nm+1)$is obtained for$N$odd and$m\in \mathbb{N}$. In particular, this gives a beautiful relation between$\unicode[STIX]{x1D701}(3),\unicode[STIX]{x1D701}(5),\unicode[STIX]{x1D701}(7),\unicode[STIX]{x1D701}(9)$and$\unicode[STIX]{x1D701}(11)$. New results involving infinite series of hyperbolic functions with$n^{2}$in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.