# Dr Geoffrey Campbell

## Areas of expertise

- Algebra And Number Theory 010101
- Combinatorics And Discrete Mathematics (Excl. Physical Combinatorics) 010104
- Real And Complex Functions (Incl. Several Variables) 010111
- Pure Mathematics Not Elsewhere Classified 010199
- Numerical And Computational Mathematics Not Elsewhere Classified 010399
- Earth Sciences Not Elsewhere Classified 049999

## Research interests

*q*series identities (Basic hypergeometric series),- Tiling patterns arising from theory of quasicrystals,
- Dirichlet series analogues of
*q*series, - Dirichlet series and Riemann zeta functions,
- Theory of Partitions,
- Arithmetical functions and divisor functions,
- Visible Point Vector (vpv) identities (includes lattice points in regions),
- Ordinary hypergeometric series,
- Number Theory in general including Diophantine equations,
- Voting Methodologies,
- Combinatorial objects in Enumerative Word problems,
- Schur functions, and the Google search algorithms.

## Researcher's projects

**I have two research monographs and quite a few number theory research papers as work in progress.**

**Monograph 1: Vector Partitions**

In my papers of the 1990s up to 2000 I introduced new combinatorial identities - these are cited in the WolframMathWorld online encyclopaedia under 'Visible Point Vector Identities'. See http://mathworld.wolfram.com/VisiblePointVectorIdentity.html. Many of the papers either appeared out of chronology or were accepted by referees but later withdrawn as there were page charges imposed. The work has progressed beyond the papers themselves and recent research on computational identities by Professor Jon Borwein at University of Newcastle can be applied to give new and interesting results. Also the identities appear to have applications in Vector Partition theory extending the works of Professor George E Andrews on integer partitions.

**Monograph 2: Dirichlet Series Analogues of q Series**

I discovered a transform that maps the classical basic hypergeometric series identities onto classes of Dirichlet summations involving Riemann zeta functions and divisor functions. In my 2006 paper (see http://www.springerlink.com/content/1nk5046814024n36/) I also discovered that the coefficients in the new identities had interpretations in terms of tiling patterns known to the theory of quasicrystals. There is a team in Germany led by Professor Michael Baake specialising in these quasicrystal tiling researches.

The theory underlying the new Dirichlet series analogues, goes into the theory of aperiodic tiling patterns such as those that occur in the seminal works of the 2011 Nobel Prize works of Professor Dan Shechtman, who discovered quasicrystals in nature. This was a paradigm shift in the science of crystallography.

An extended list of new Dirichlet series analogue identities resultant from the transform has not yet been published, but will be included in this monograph. The quasicrystal tilings that enumerate coefficients in the new Dirichlet series analogues of the q series are dependent on the theory underlying Coincidence Site Lattices. My colloquium in 2011 highlighted some of these tiling patterns arising from the Dirichlet series analogues of the basic hypergeometric sums. (see http://www.amsi.org.au/index.php/events-mainmenu/agr-events/737-drichlet-analogues-of-q-series-and-some-tiling-patterns-from-quasicrystals)

Features of this are:-

*Note: I gave colloquia at LaTrobe University on these two monograph subject areas in March and September 2011.*

**Papers arising from LinkedIn Number Theory Group**

As manager of the online LinkedIn Number Theory Group, I have about 30 papers arising from discussions in that forum, many papers of which I am co-authoring with members of the group. As these papers are drafted and settle, they will appear below with a link to the arxive or to the Journal or other publication point.

## Publications

- CAMPBELL G.B. The q-Dixon sum Dirichlet series analogue, http://arxiv.org/abs/1302.2664, 2013.
- CAMPBELL G B, Polylogarithm approaches to Riemann Zeta function zeroes; http://arxiv.org/abs/1212.2246; December 2012.
- CAMPBELL, G.B. Ramanujan and Eckford Cohen totients from Visible Point Identities, http://arxiv.org/abs/1212.2818; December 2012
- CAMPBELL, G. B. Dirichlet series analogues of q-shifted factorial and the q-Kummer sum, http://arxiv.org/abs/1212.2248; December 2012.
- CAMPBELL, G. B. Generating Functions for Vector Partitions, Submitted 2011 to Int J Number Theory.
- Brainsmatter Podcast No 61: "Visible Lattice Points" interview with Dr Geoffrey Campbell, Honorary research Fellow, La Trobe University, aired Sunday 27 April 2008. (downloaded from www.brainsmatter.com/?p=176 over 500 times)
- Brainsmatter Podcast No 58: "The genius of Srinivasa Ramanujan" interview with Dr Geoffrey Campbell, Honorary research Fellow, La Trobe University, aired Tuesday 18 March 2008. (downloaded from www.brainsmatter.com/?p=174 over 7000 times)
- CAMPBELL, G. B. An Euler Product transform applied to q series, Ramanujan J (2006) 12:267-293.
- CAMPBELL, G. B. Characterization of three dimensional images by visible point vector projections, unpublished paper, 2001. (The author liaised with the Supercomputer Lab staff at ANU in 2000 in order to establish a perspective on the value of the ideas underlying this work.)
- CAMPBELL, G. B. A generalized Ramanujan arithmetical function, and Jordan's totient, Internat. J. Math. & Math. Sci., (was to appear 2000 but was withdrawn by the journal due to page charges)
- CAMPBELL, G. B. Combinatorial identities related to vector partitions, Internat. J. of Math, & Math. Sci., (was to appear 2000 but was withdrawn due to page charges)
- CAMPBELL, G. B. Dirichlet series analogues of bilateral q-series, Internat. J. of Math, & Math. Sci., (was to appear 2000 but was withdrawn due to page charges)
- CAMPBELL, G. B. Dirichlet series analogues of q-series, Internat. J. of Math, & Math. Sci., (was to appear 2000 but was withdrawn due to page charges)
- CAMPBELL, G. B. Dirichlet series related D-analogues of the q-gamma and q-beta functions, Internat. J. Math. & Math. Sci., (was to appear 2000 but was withdrawn due to page charges)
- CAMPBELL, G. B. Further cases of Dirichlet series analogues of q-series, Internat. J. of Math, & Math. Sci., (was to appear 2000 but was withdrawn due to page charges)
- CAMPBELL, G. B. Further Dirichlet series analogues of q-series, Internat. J. of Math, & Math. Sci., (was to appear 2000 but was withdrawn due to page charges)
- CAMPBELL, G. B. Infinite products over hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 23, No 4, 2000, 271-277.
- CAMPBELL, G. B. Partial Euler Product Dirichlet series analogues of q-series, Internat. J. of Math, & Math. Sci., (was to appear 2000 but was withdrawn due to page charges)
- CAMPBELL, G. B. Dirichlet series analogues of q-series, Mathematics Research Report No 2000.014, (this was a 59 page portion of the manuscript submitted in 1999 as a monograph to Lecture Notes in Mathematics, Springer Verlag, New York The manuscript went missing at Springer Verlag along with the one cited above at 20, but is to be resubmitted)
- CAMPBELL, G. B. A closer look at some new identities, Internat. J. Math. & Math. Sci., Vol 21, No 3, 1998, pp581-586.
- CAMPBELL, G. B. A New Class of Identities akin to q-Series in Several Variables, Research paper no (to be determined), Centre for Mathematics and its applications, The Australian National University, 1998
- CAMPBELL, G. B. Combinatorial Identities in Number Theory related to q-series and Arithmetical functions, Bull. Austral. Math. Soc., Vol. 58, (1998) pp345-347.
- CAMPBELL, G. B. Combinatorial Infinite Products in Several Variables over Visible Lattice Points, Lecture Notes in Mathematics, Springer Verlag, New York, (Submitted 1998 as a monograph, but misplaced by the editors. An updated version of 160 pages in length has been resubmitted.)
- CAMPBELL, G. B. Visible point vector summations from hypercube and hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 21, No 4, 741-748, 1998.
- CAMPBELL, G. B. On generating functions for vector partitions, Research paper no 55-97, Centre for Mathematics and its applications, The Australian National University, 1997.
- CAMPBELL, G. B. A generalised formula of Hardy, Int. J. Math. Math. Sci., Vol 17, No 2, 1994, 369-378.
- CAMPBELL, G. B. A new class of infinite product, and Euler's totient, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 417-422.
- CAMPBELL, G. B. Infinite products over visible lattice points, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 637-654.
- CAMPBELL, G. B. Dirichlet summations and products over primes, Internat. J. Math. & Math. Sci., Vol 16, No 2, 1993, 359-372.
- CAMPBELL, G. B. Formulae with functions exhibiting self-similarity, Research Paper preprint series, Centre for Mathematics and its Applications, The Australian National University, 1993.
- CAMPBELL, G. B. Multiplicative functions over Riemann zeta function products, J. Ramanujan Soc. 7 No. 1, 1992, 52-63.