Dedekind Eta Function Research Articles

BILATERAL ZETA FUNCTIONS AND THEIR APPLICATIONS

We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta function. Especially, we prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, We obtain simple proofs of some formulas, for example the reflection formula for the multiple gamma function, the inversion formula of the Dedekind eta function, Ramanujan's formula, Fourier expansion of the Barnes zeta function and multiple Iseki's formula.

EXPLICIT FORMULAS FOR THE FOURIER COEFFICIENTS OF A CLASS OF ETA QUOTIENTS

Recently, Williams discovered explicit formulas of the coefficients c(n) in the Fourier series expansions of a class of eta quotients. Motivated by the results obtained by Williams, we find that the coefficients c(2n) in the Fourier series expansions of another class of eta quotients can be represented as a linear combination of σ3(n), σ3(n/2), σ3(n/3) and σ3(n/6). One example is [Formula: see text] where [Formula: see text]

Stirling modular forms and special values of multiple cotangent functions

We give a new proof of the modularity of the Dedekind eta function using a transformation formula of Stirling modular forms. Moreover, we study special values of multiple cotangent functions. Then we obtain some relations between special values of multiple cotangent functions.

FOURIER SERIES OF A CLASS OF ETA QUOTIENTS

The sum of divisors function σ(m) is defined by [Formula: see text] Let [Formula: see text] denote the upper half of the complex plane. Let η(z)[Formula: see text] be the Dedekind eta function. A class [Formula: see text] of eta quotients is given for which the Fourier series of each member of [Formula: see text] can be given explicitly. One example is [Formula: see text] where [Formula: see text]

INFINITE SERIES RELATION FROM A MODULAR TRANSFORMATION FORMULA FOR THE GENERALIZED EISENSTEIN SERIES

In 1970s, B. C. Berndt proved a transformation formula for a large class of functions that includes the classical Dedekind eta function. From this formula, he evaluated several classes of infinite series and found a lot of interesting infinite series identities. In this paper, using his formula, we find new infinite series identities.

Congruences for Andrews’ spt-function modulo powers of $5$, $7$ and $13$

Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo p for all primes p>3 which were conjectured earlier by the author. We extend Ono's method to handle the powers of 5, 7 and 13 congruences. We need the theory of weak Maass forms as well as certain classical modular equations for the Dedekind eta-function.

POWERS OF THE ETA-FUNCTION AND HECKE OPERATORS

Half-integer weight Hecke operators and their distinct properties play a major role in the theory surrounding partition numbers and Dedekind's eta-function. Generalizing the work of Ono in [K. Ono, The partition function and Hecke operators, Adv. Math.228 (2011) 527–534], here we obtain closed formulas for the Hecke images of all negative powers of the eta-function. These formulas are generated through the use of Faber polynomials. In addition, congruences for a large class of powers of Ramanujan's Delta-function are obtained in a corollary. We further exhibit a fast calculation for many large values of vector partition functions.

BKM superalgebras from counting dyons in [formula omitted] supersymmetric type II compactifications

We study the degeneracy of quarter BPS dyons in N=4 type II compactifications of string theory. We find that the genus-two Siegel modular forms generating the degeneracies of the quarter BPS dyons in the type II theories can be expressed in terms of the genus-two Siegel modular forms generating the degeneracies of quarter BPS dyons in the CHL theories and the heterotic string. This helps us in understanding the algebra structure underlying the degeneracy of the quarter BPS states. The Conway group, Co0, plays a role similar to Mathieu group, M24, in the CHL models with eta quotients appearing in the place of eta products. We construct BKM Lie superalgebra structures for the ZN (for N=2,3,4) orbifolds of the type II string compactified on a six-torus.

Algebraic integers as special values of modular units

AbstractLet $\varphi(\tau)=\eta(\tfrac12(\tau+1))^2/\sqrt{2\pi}\exp\{\tfrac14\pi\ri\}\eta(\tau+1)$, where η(τ) is the Dedekind eta function. We show that if τ0 is an imaginary quadratic argument and m is an odd integer, then $\sqrt{m}\varphi(m\tau_0)/\varphi(\tau_0)$ is an algebraic integer dividing $\sqrt{m}$ This is a generalization of a result of Berndt, Chan and Zhang. On the other hand, when K is an imaginary quadratic field and θK is an element of K with Im(θK) > 0 which generates the ring of integers of K over ℤ, we find a sufficient condition on m which ensures that $\sqrt{m}\varphi(m\theta_K)/\varphi(\theta_K)$ is a unit.

Resurgence of the Kontsevich-Zagier series

We give an explicit formula for the Borel transform of the power series when q=e 1/x from which its analytic continuation, its singularities (all on the positive real axis) and the local monodromy can be manifestly determined. We also give two formulas (one involving the Dedekind eta function, and another involving the complex error function) for the right, left and median summation of the Borel transform. We also prove that the limiting values of the median sum at rational multiples of 1/(2πi) coincide with the values of f(q) at the corresponding complex roots of unity. Our resurgence theorem extends more generally to the power series of torus knots and Seifert fibered 3-manifolds associated by Quantum Topology.

Weight of a link in a shortest path tree and the Dedekind Eta function

AbstractThe weight of a randomly chosen link in the shortest path tree on the complete graph with exponential i.i.d. link weights is studied. The corresponding exact probability generating function and the asymptotic law are derived. As a remarkable coincidence, this asymptotic law is precisely the same as the distribution of the cost of one “job” in the random assignment problem. We also show that the asymptotic (scaled) maximum interattachment time to that shortest path tree, which is a uniform recursive tree, equals the square of the Dedekind Eta function, a central function in modular forms, elliptic functions, and the theory of partitions. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

CONSTRUCTION OF EISENSTEIN SERIES FOR Γ0(p)

A simple construction of Eisenstein series for the congruence subgroup Γ0(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty.

Modularity of Calabi-Yau varieties and conformal field theory

We study the modularity problem of Calabi-Yau varieties from the conformal field theoretic point of view. We express the modular forms associated to all 1-dimensional Calabi-Yau orbifolds in terms of products of Dedekind eta functions, which is hoped to shed light on the modularity questions for higher dimensional Calabi-Yau varieties.

A Transformation Law of an Eta Product and the Invariance of a Class of Entire Modular Functions Under $\Gamma _0 (n)$

We present a new proof, using Residue Calculus, of the transformation law of a general eta product under $\Gamma_0(n)$ where $n$ is any integer, then we deduce the invariance of a special case of this product under this group and we prove the transformation law of another special case. Our proof is inspired by Siegel's proof [7] of the transformation law of the Dedekind eta function and by Rademacher's generalization [5].

Quotients of values of the Dedekind Eta function

Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form $${\frac{\eta(A_jz)} {\eta(A_{j-1}z)}}$$ , where A j-1 and A j are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation XΩ of a fixed region Ω in $${\mathbb{R}^2}$$ , and z is a fixed complex number in the upper half plane. In particular, we show that the limiting distribution of these quotients depends heavily on the index of Farey fractions which was first introduced and studied by Hall and Shiu. The distribution of Farey fractions with respect to the value of the index dictates the universal limiting behavior of these quotients. Motivated by chains of these quotients, we show how to obtain a generalization, due to Zagier, of an important formula of Hall and Shiu on the sum of the index of Farey fractions.

Schläfli modular equations for generalized Weber functions

Sets of appropriately normalized eta quotients, that we call level n Weber functions, are defined, and certain identities generalizing Weber function identities are proved for these functions. Schläfli type modular equations are explicitly obtained for Generalized Weber Functions associated with a Fricke group Γ0(n)+, for n=2,3,5,7,11,13 and 17.

A new proof of the transformation law of Jacobi’s theta function $\theta _3(w,\tau )$

. We present a new proof, using Residue Calculus, of the transformation law of the Jacobi theta function θ 3 (ω,τ) defined in the upper half plane. Our proof is inspired by Siegel's proof of the transformation law of the Dedekind eta function.

An identity for the Dedekind eta-function involving two independent complex variables

The authors prove a new identity for the Dedekind eta-function that involves third powers of the eta-function, with each of the two cubes being a function of a different complex variable.

Ramanujan's Eisenstein series and powers of Dedekind's eta-function

In this article, we use the theory of elliptic functions to construct theta function identities which are equivalent to Macdonald's identities for A2, B2 and G2. Using these identities, we express, for d = 8, 10 or 14, certain theta functions in the form d()F(P, Q, R), where () is Dedekind's eta-function, and F(P, Q, R) is a polynomial in Ramanujan's Eisenstein series P, Q and R. We also derive identities in the case when d = 26. These lead to a new expression for 26(). This work generalizes the results for d = 1 and d = 3 which were given by Ramanujan on page 369 of ?The Lost Notebook?

Characters, supercharacters and Weber modular functions

We study certain canonical automorphic forms associated to irreducible characters of N = 1 superconformal minimal models in both the Neveu-Schwarz and Ramond sector by using representation theoretic methods. By extending the techniques from [A. Milas, Virasoro algebra, Dedekind eta-function and Specialized Macdonald's identities, Transf. Groups 9 (2004), 273–288.] to the setting of N = 1 superconformal algebras we obtain a series of modular identities involving certain Wronskian determinants in both Neveu-Schwarz (untwisted) and Ramond (twisted) sector. Three Weber modular functions play a fundamental role here as well as the Dedekind η-function. In the most interesting case of minimal series, we obtain a derivation and generalizations of several classical modular q-series identities (e.g., Jacobi's Four Square Theorem, a Carlitz' identity, specialized Macdonald's identities for B series, etc.)