Searcharxiv⌕ Search

arXiv subjects

Robert Osburn

Publications and source records attributed to Robert Osburn.

At least 19 recordsLinked to original sources
arXiv2026

Bailey pairs, Eichler integrals and unified Witten-Reshetikhin-Turaev invariants

In 1999, Lawrence and Zagier expressed the Witten-Reshetikhin-Turaev (WRT) invariant of the Poincar\'e homology sphere as the limiting value of the Eichler integral of a weight 3/2 modular form. Habiro's construction of the unified WRT invariant subsequently recast this result as an identity for a $q$-hypergeometric series at roots of unity. This motivated Hikami to prove analogous $q$-series identities involving the unified WRT invariants of certain Brieskorn homology spheres. Hikami also made several conjectures of a similar type for $q$-series with no apparent connection to quantum invariants. In this paper we use the Bailey pair machinery and a novel relation between incomplete quadratic Gauss sums with periodic coefficients to construct infinite families of identities between $q$-multisums at roots of unity and limiting values of Eichler integrals of weight 3/2 modular forms. These identities include all of Hikami's results and conjectures as well as a generalization of the result of Lawrence and Zagier.

math.NT↗
arXiv2025

Generalized rank deviations for overpartitions

We prove formulas for generalized rank deviations for overpartitions. These formulas are in terms of Appell-Lerch series and sums of quotients of theta functions and extend work of Lovejoy and the second author. As an application, we compute a dissection.

math.NT↗
arXiv2025

Arborescent links and modular tails

We prove an explicit formula for the tail of the colored Jones polynomial for a class of arborescent links in terms of a product of theta functions and/or false theta functions. We also provide numerical evidence towards a classification of the modularity of tails of the colored Jones polynomial for alternating knots.

math.GT↗
arXiv2024

Unimodal sequences and mixed false theta functions

We consider two-parameter generalizations of Hecke-Appell type expansions for the generating functions of unimodal and special unimodal sequences. We then determine their explicit representations which involve mixed false theta functions. These results complement recent striking work of Mortenson and Zwegers on the mixed mock modularity of the generalized $U$-function due to Hikami and Lovejoy. As an application, we demonstrate how to recover classical partial theta function identities which appear in Ramanujan's lost notebook and in work of Warnaar.

math.NT↗
arXiv2023

On sporadic sequences

In this note, we prove the last remaining case of the original 15 two-term supercongruence conjectures for sporadic sequences. The proof utilizes a new representation for this sequence (due to Gorodetsky) as the constant term of powers of a Laurent polynomial.

math.NT↗
arXiv2023

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.

math.NT↗
arXiv2023

Resurgence, Habiro elements and strange identities

We prove resurgence properties for the Borel transform of a formal power series associated to elements in the Habiro ring that come from radial limits of partial theta series via strange identities. As an application, we prove a conjecture in quantum topology due to Costin and Garoufalidis for two families of torus knots.

math.NT↗
arXiv2022

Asymptotics and sign patterns for coefficients in expansions of Habiro elements

We prove asymptotics and study sign patterns for coefficients in expansions of elements in the Habiro ring which satisfy a strange identity. As an application, we prove asymptotics and discuss positivity for the generalized Fishburn numbers which arise from the Kontsevich-Zagier series associated to the colored Jones polynomial for a family of torus knots. This extends Zagier's result on asymptotics for the Fishburn numbers.

math.NT↗
arXiv2020

Quantum modularity of partial theta series with periodic coefficients

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series $\mathscr{F}_t(q)$ which matches (at a root of unity) the colored Jones polynomial for the family of torus knots $T(3,2^t)$, $t \geq 2$, is a weight $3/2$ quantum modular form. This generalizes Zagier's result on the quantum modularity for the "strange" series $F(q)$.

math.NT↗
arXiv2020

Variations of Andrews-Beck type congruences

We prove three variations of recent results due to Andrews on congruences for $NT(m,k,n)$, the total number of parts in the partitions of $n$ with rank congruent to $m$ modulo $k$. We also conjecture new congruences and relations for $NT(m,k,n)$ and for a related crank-type function.

math.NT↗
arXiv2020

Generalized Fishburn numbers and torus knots

Andrews and Sellers recently initiated the study of arithmetic properties of Fishburn numbers. In this paper, we prove prime power congruences for generalized Fishburn numbers. These numbers are the coefficients in the $1-q$ expansion of the Kontsevich-Zagier series $\mathscr{F}_{t}(q)$ for the torus knots $T(3,2^t)$, $t \geq 2$. The proof uses a strong divisibility result of Ahlgren, Kim and Lovejoy and a new "strange identity" for $\mathscr{F}_{t}(q)$.

math.NT↗
arXiv2019

The colored Jones polynomial and Kontsevich-Zagier series for double twist knots, II

Let $K_{(m,p)}$ denote the family of double twist knots where $2m-1$ and $2p$ are non-zero integers denoting the number of half-twists in each region. Using a result of Takata, we prove a formula for the colored Jones polynomial of $K_{(-m,-p)}$ and $K_{(-m,p)}$. The latter case leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. We also use Bailey pairs and formulas of Walsh to find cyclotomic-like expansions for the colored Jones polynomials of $K_{(m,p)}$ and $K_{(m,-p)}$.

math.GT↗
arXiv2018

Constrained ternary integers

An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the $n^{th}$ (inverse) cyclotomic polynomials with ternary $n$. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction $\ge 0.925$ of ternary integers.

math.NT↗
arXiv2018

Interpolated sequences and critical $L$-values of modular forms

Recently, Zagier expressed an interpolated version of the Apéry numbers for $ζ(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier's six sporadic sequences are essentially critical $L$-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown's cellular integrals and critical $L$-values of modular forms of odd weight.

math.NT↗
arXiv2018

Sequences, modular forms and cellular integrals

It is well-known that the Apéry sequences which arise in the irrationality proofs for $ζ(2)$ and $ζ(3)$ satisfy many intriguing arithmetic properties and are related to the $p$th Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

math.NT↗
arXiv2017

A modular supercongruence for $_6F_5$: an Apéry-like story

We prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated ${}_6F_5$-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to $ζ(3)$ to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence between the Apéry numbers and another Apéry-like sequence.

math.NT↗
arXiv2017

The colored Jones polynomial and Kontsevich-Zagier series for double twist knots

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots $K_{(-m,-p)}$ and $K_{(-m,p)}$ where $m$ and $p$ are positive integers. In the $(-m,-p)$ case, this leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of $K_{(m,p)}$ gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function for strongly unimodal sequences.

math.GT↗