Robert Osburn
University College Dublin
Network
Latest external collaboration on country level. Dive into details by clicking on the dots.
Publication
Featured researches published by Robert Osburn.
Mathematics of Computation | 2009
Robert Osburn; Carsten Schneider
Let p be an odd prime. In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have been of interest as they are related to the number of F_p points on algebraic varieties and to Fourier coefficients of modular forms. In this paper, we explicitly determine these functions modulo higher powers of p and discuss an application to supercongruences. This application uses two non-trivial generalized Harmonic sum identities discovered using the computer summation package Sigma. We illustrate the usage of Sigma in the discovery and proof of these two identities.
Acta Arithmetica | 2007
Jeremy Lovejoy; Robert Osburn
In 1954, Atkin and Swinnerton-Dyer proved Dysons conjectures on the rank of a partition by establishing formulas for the generating functions for rank differences in arithmetic progressions. In this paper, we prove formulas for the generating functions for rank differences for overpartitions. These are in terms of modular functions and generalized Lambert series.
Integers | 2011
Jeremy Lovejoy; Robert Osburn
Abstract Recently, Andrews, Hirschhorn and Sellers have proven congruences modulo 3 for four types of partitions using elementary series manipulations. In this paper, we generalize their congruences using arithmetic properties of certain quadratic forms.
arXiv: Number Theory | 2016
Robert Osburn; Brundaban Sahu; Armin Straub
We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss re- cent progress and future directions concerning other types of supercon- gruences.
Journal of Mathematical Analysis and Applications | 2016
Robert Osburn; Wadim Zudilin
Abstract We prove the last remaining case of the original 13 Ramanujan-type supercongruence conjectures due to Van Hamme from 1997. The proof utilizes classical congruences and a WZ pair due to Guillera. Additionally, we mention some future directions concerning this type of supercongruence.
arXiv: Number Theory | 2010
Robert Osburn; Brundaban Sahu
We prove two congruences for the coecients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Ap ery-like dierential equations.
Journal of Number Theory | 2003
Robert Osburn; Brian Murray
Abstract There has been recent progress on computing the 4-rank of the tame kernel K 2 ( O F ) for F a quadratic number field. For certain quadratic number fields, this progress has led to “density results” concerning the 4-rank of tame kernels. These results were first mentioned in Conner and Hurrelbrink (J. Number Theory 88 (2001) 263) and proven in Osburn (Acta Arith. 102 (2002) 45). In this paper, we consider some additional quadratic number fields and obtain further density results of 4-ranks of tame kernels. Additionally, we give tables which might indicate densities in some generality.
International Journal of Number Theory | 2005
Stephen Choi; Angel V. Kumchev; Robert Osburn
Let r3(n) be the number of representations of a positive integer n as a sum of three squares of integers. We give two alternative proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of r3(n).
Glasgow Mathematical Journal | 2017
Jeremy Lovejoy; Robert Osburn
We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujans seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.
arXiv: Number Theory | 2005
Robert Osburn
We prove the remaining case of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n = x 2 + Ny 2 for a squarefree integer N.