Theta Functions Research Articles

Probabilistic aspects of Jacobi theta functions

In this note we deduce well-known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and $+1$, (ii) killing at $-1$ and $+1$. It is seen that these two representations give, in a sense, most compact forms of the modular theta-function identities. We study also discrete Gaussian distributions generated by theta functions, and derive, in particular, addition formulas for discrete Gaussian variables.

Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts

We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and the first author in 1988. As a consequence, we give new versions of their definitions without adjusted weights. Furthermore, we investigate the combinatorial interpretation of an $M_2$-crank of partitions without repeated odd parts and explore connections of these statistics with their companion rank counterparts and the tenth order mock theta functions of Ramanujan.

Witten–Reshetikhin–Turaev invariants and indefinite false theta functions for plumbing indefinite H-graphs

Gukov–Pei–Putrov–Vafa conjectured the existence of [Formula: see text]-series whose radial limits are Witten–Reshetikhin–Turaev invariants and called them homological blocks. For weakly negative definite plumbed 3-manifolds, Gukov–Pei–Putrov–Vafa and Gukov–Manolescu constructed homological blocks. In this paper, we construct indefinite false theta functions which are candidates of homological blocks for some plumbed 3-manifolds which are not weakly negative definite. Moreover we prove that, for the Poincaré homology sphere, our indefinite false theta function coincides with the original homological block.

Proofs of three conjectured internal congruences modulo 32 for Schur-type overpartitions

Let S(n) denote the number of overpartitions of Schur-type. Recently, Chern, da Silva and Sellers proved many congruences modulo 8 and 16 for S(n). At the end of their paper, they also posed a conjecture on internal congruences modulo 32 for S(n). In this paper, we confirm their conjecture by using theta function identities and the (p, k)-parameterization of theta functions given by Alaca and Williams. In particular, we show that for any integer j with 0 ≤ j ≤ 31, there are infinitely many integers uj such that S(uj ) ≡ j (mod 32).

Polynomial Analogs of Theta Functions and Rational Solutions of the KP Equation

Abstract In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.

Fredholm Determinant and Wronskian Representations of the Solutions to the Schrödinger Equation with a KdV-Potential

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and Fredholm determinants and we obtain multi-parametric solutions to this equation. As a consequence, a double Wronskian representation of the solutions to this equation is constructed. We also give quasi-rational solutions to this Schrödinger equation with rational KdV potentials.

Dark solitons on elliptic function background for the defocusing Hirota equation

Abstract We investigate dark solitons lying on elliptic function background in the defocusing Hirota equation with third-order dispersion and self-steepening terms. By means of the modified squared wavefunction method, we obtain the Jacobi’s elliptic solution of the defocusing Hirota equation, and solve the related linear matrix eigenvalue problem on elliptic function background. The elliptic N-dark soliton solution in terms of theta functions is constructed by the Darboux transformation and limit technique. The asymptotic dynamical
behaviors for the elliptic N-dark soliton solution as t → ±∞ are studied. Through numerical plots of the elliptic one-, two- and three-dark solitons, the amplification effect on the velocity of elliptic dark solitons, and the compression effect on the soliton spatiotemporal distributions produced by the third-order dispersion and self-steepening terms are discussed.

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.

Congruences between the coefficients of certain mock theta functions

In this article, we prove several congruences modulo 3, 4, 5, 8, 9, 12, 24, 27, 81, 243, and 729 enjoyed by the coefficients of certain mock theta functions. As an example, for the second order mock theta functionsμ2(q):=∑n=0∞(−1)nqn2(q;q2)n(−q2;q2)n2=∑n=0∞Pμ2(n)qn,B2(q):=∑n=0∞qn(n+1)(−q2;q2)n(q;q2)n+12=∑n=0∞qn(−q;q2)n(q;q2)n+1=∑n=0∞PB2(n)qn, we havePμ2(27n+26)≡25PB2(108n+103)(mod27) for all n≥0.

Strange duality at level one for alternating vector bundles

In this paper, we show a strange duality isomorphism at level one for the space of generalized theta functions on the moduli spaces of alternating anti-invariant vector bundles in the ramified case. These anti-invariant vector bundles constitute one of the non-trivial examples of parahoric G-torsors, where G is a twisted (not generically split) parahoric group scheme.

A Selberg identity for the Shimura lift

AbstractWe first prove a Selberg‐type identity for the Shimura lift of the Rankin–Cohen bracket of a normalized Hecke eigenform and the theta function. We then discuss its relationship with the nonvanishing of central values of ‐functions associated to Hecke eigenforms.

Free fermions, neutrality and modular transformations

Abstract With a view towards higher-spin applications, we study the partition function of a free complex fermion in 2d conformal field theory, restricted to the neutral (zero fermion number) sector. This restriction leads to a partial theta function with a combinatoric interpretation in terms of Dyson’s crank of a partition. More crucially, this partition function can be expressed in terms of a q-hypergeometric function with quantum modular properties. This allows us to find its high-temperature asymptotics, including subleading terms which agree with, but also go beyond, what one obtains by imposing neutrality thermodynamically through a chemical potential. We evaluate the asymptotic density of states for this neutral partition function, including the first few subleading terms. Our results should be extendable to more fermions, as well as to higher-spin chemical potentials, which would be of relevance to the higher-spin/minimal model correspondence.

Irrational Outputs from Rational Inputs in Theta-3 Functions Using Dirichlet’s Theorem

Theta Functions are not an easy subject of mathematics because new researches about this wide topic are still being published regularly. The findings about Theta Functions lead directly to many applications in different fields of applied mathematics and help also to develop old interesting mathematical topics such as Poisson summation and Riemann Zeta Function. This work proves by using a theorem of Dirichlet and a previously demonstrated useful approximation that the rational inputs of Theta-3 Functions give only irrationals as outputs of these functions. This paper applies only simple notions of sequences and represents an opportunity for the students of mathematics to easily understand the steps of the demonstrations.

Ramanujan's theta functions and internal congruences modulo arbitrary powers of 3

In this work, we investigate internal congruences modulo arbitrary powers of 3 for two functions arising from Ramanujan's classical theta functions φ(q) and ψ(q). By letting∑n≥0ph3(n)qn:=φ(−q3)φ(−q)and∑n≥0ps3(n)qn:=ψ(q3)ψ(q), we prove that for any m≥1 and n≥0,ph3(32m−1n)≡ph3(32m+1n)(mod3m+2), andps3(32m−1n+32m−14)≡ps3(32m+1n+32m+2−14)(mod3m+2), thereby substantially generalizing the previous results of Bharadwaj et al. and Gireesh et al., respectively.

On the Approximation of the Hardy Z-Function via High-Order Sections

The Z-function is the real function given by Z(t)=eiθ(t)ζ12+it, where ζ(s) is the Riemann zeta function, and θ(t) is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has traditionally posed significant computational challenges. This research addresses these challenges by exploring new methods for approximating Z(t) and its zeros. The sections of Z(t) are given by ZN(t):=∑k=1Ncos(θ(t)−ln(k)t)k for any N∈N. Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE) Z(t)≈2ZN˜(t)(t) for N˜(t)=t2π. While historically important, the Hardy–Littlewood AFE does not sufficiently discern the RH and requires further evaluation of the Riemann–Siegel formula. An alternative, less common, is Z(t)≈ZN(t)(t) for N(t)=t2, which is Spira’s approximation using higher-order sections. Spira conjectured, based on experimental observations, that this approximation satisfies the RH in the sense that all of its zeros are real. We present a proof of Spira’s conjecture using a new approximate equation with exponentially decaying error, recently developed by us via new techniques of acceleration of series. This establishes that higher-order approximations do not need further Riemann–Siegel type corrections, as in the classical case, enabling new theoretical methods for studying the zeros of zeta beyond numerics.

Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant Γ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma =2$$\\end{document} and that the number n of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order 1/n from the hard edge. At distances much larger than 1/n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/\\sqrt{n}$$\\end{document}, the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the “semi-hard edge”. More precisely, we provide asymptotics for the correlation kernel Kn(z,w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{n}(z,w)$$\\end{document} as n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document} in two microscopic regimes (with either |z-w|=O(1/n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|z-w| = \\mathcal{O}(1/\\sqrt{n})$$\\end{document} or |z-w|=O(1/n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|z-w| = \\mathcal{O}(1/n)$$\\end{document}), as well as in three macroscopic regimes (with |z-w|≍1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|z-w| \\asymp 1$$\\end{document}). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels.

A numerical representation of hyperelliptic KdV solutions

The periodic and quasi-periodic solutions of the integrable system have been studied for four decades based on the Riemann theta functions. However, there is a fundamental difficulty in representing the solutions numerically because the Riemann theta function requires several transcendental parameters. This paper presents a novel method for the numerical representation of such solutions from the algebraic treatment of the periodic and quasi-periodic solutions of the Baker–Weierstrass hyperelliptic ℘ functions. We demonstrate the numerical representation of the hyperelliptic ℘ functions of genus two.

Hypergeometric Function and its Application on Heat Kernel in a Torus Surface with a Rectangular Lattice

Hypergeometric functions are transcendental functions that are applicable in various branches of mathematics, physics, and engineering. They are solutions to a class of differential equations called hypergeometric differential equations. In this paper we will be using theta functions, null zeta function and Ramanujan’s identities to study the behavior of heat kernel in a torus surface with rectangular Lattice.

Rank deviations for overpartitions

We prove general fomulas for the deviations of two overpartition ranks from the average. These formulas are in terms of Appell–Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the numerous overpartition rank difference identities in the literature. We give two illustrations.

Adiabatic Limit, Theta Function, and Geometric Quantization

Adiabatic Limit, Theta Function, and Geometric Quantization