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Dive into the research topics where Robert Osburn is active.

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Featured researches published by Robert Osburn.


Mathematics of Computation | 2009

GAUSSIAN HYPERGEOMETRIC SERIES AND SUPERCONGRUENCES

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

RANK DIFFERENCES FOR OVERPARTITIONS

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

Quadratic forms and four partition functions modulo 3

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

Supercongruences for sporadic sequences

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

On the (K.2) supercongruence of Van Hamme

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

Congruences via modular forms

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

Tame kernels and further 4-rank densities

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

ON SUMS OF THREE SQUARES

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

MOCK THETA DOUBLE SUMS

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

A remark on a conjecture of Borwein and Choi

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.

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Brundaban Sahu

National Institute of Science Education and Research

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Song Heng Chan

Nanyang Technological University

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David Brink

University College Dublin

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Brian Murray

Louisiana State University

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