# Geoffrey Campbell

## Contacts

### Research interests

Recently I was contracted to write a book monograph on ** Vector Partitions and the Visible Points** for Chapman and Hall Publishers. My motivation is to cover classical Integer Partitions, partitions in Statistical Mechanics Solved Models such as the Hard Hexagon, and bring this to a research level with the

*Visible Point Vector identities*I found some years ago.

Along the way, we meet new relationships and identities involving Plane Partitions, Ono's exact Partition Formula, Polylogarithms, generalized Euler Sums and Mordell-Wittam-Tornheim ensembles. I expect the book to run to 300 to 400 pages and to be completed well prior to June 2022.

My high level table of contents, based on my draft pages to January 2021, gives you a sense of where the book is heading:

Preface

Chapter 1. Historical background - The range of literature

1. The G. E. Andrews book and his book co-authored with K. Eriksson

2. Basic Hypergeometric Series

3. Lattice Sums in Chemistry molecular structures

4. Polylogarithms and computational research related results

5. Partition theory in Statistical Mechanics and Theoretical Physics

Chapter 2. General introduction and context for partitions

Chapter 3. Integer partitions and their generating functions

Chapter 4. Plane partitions

Chapter 5. Asymptotic formulas for integer partitions

Chapter 6. The partition function in Statistical Mechanics

Chapter 7. Vector partitions and their generating function identities

Chapter 8. Integer Partitions to Vector Partitions

Chapter 9. Weighted Vector Partitions as hybrid n-space variations

Chapter 10. Functional Equations for n-space Vector Partitions

Chapter 11. Visible Point Vector Identities in the first Hyperquadrant

Chapter 12. Visible Point Vector Identities in Hyperpyramid lattices

Chapter 13. Visible Point Vector identities related to particular Euler sum values

Chapter 14. Polylogarithms, generalized Euler Sums and Mordell-Wittam-Tornheim functions

Chapter 15. Visible Point Vector Identities in Skewed Hyperpyramid lattices

Chapter 16. The Ramanujan trigonometric function and visible point identities

Chapter 17. Other Non-weighted n-space Vector Partition Theorems

Chapter 18. Determinants, Bell Polynomial Expansions for Vector Partitions

Chapter 19. The 2D and 3D Light Diffusion Models

Chapter 20. Partition Grids for unweighted 2D VPVs II

Chapter 21. Partition Grids for unweighted 2D VPVs III

Chapter 22. Partition Grids for weighted 2D VPVs IV

Chapter 23. The 3D Light Diffusion Model in the first hyperquadrant.

Bibliography

In 2006 I published an introductory paper for ** Dirichlet series analogues of q-series**, which led me to arithmetical function identities encoding so-named quasicrystals. This area of research is still being developed, and I am drafting a monograph on this topic as well.

I therefore can say I am interested in the Theory of Higher Dimensional Partitions, Aperiodic Order, Dynamical Systems, Combinatorics, Discrete Geometry, Number Theory, Quasicrystal tilings and their Dirichlet series functions, and Mathematical Physics, and areas where these theories may overlap.

In 2019 I wrote a paper generalizing the *q*-binomial theorem into a Euclidean *n*-space formula with a view to formalising an approach to vector partition theory. (See https://arxiv.org/abs/1906.07526) The substance of an updated version of this paper has led to my writing the above detailed book monograph project on *Vector Partitions and the Visible Points*.

I am also Manager of the almost 40K member ** LinkedIn Number Theory Group** located at https://www.linkedin.com/groups/4510047/. I have posted many brief mathematical problems and news stories at graduate and research levels over the past seven years in that forum. An informative post in that group is http://bit.ly/3a6Wyhc.

I am also an Administrator for the Facebook Group ** Classical Mathematics** presently having about 4,500 members. A typical post in that group is http://bit.ly/2NFtbLn.

### Groups

- Collaborator, Mathematical physics

**I am a published poet as well as a mathematics person. **

**POEMS: **

In 2010 I published *Words in Common*, which is a collection over decades of *journal/magazine/anthology* published poems partly funded by the Australia Council for the Arts many years ago. The book was officially launched by Professor Kevin Brophy from University of Melbourne, Creative Writing Department. It was edited by Associate Professor Trevor Code from Deakin University.

See a link with excerpts from the poems at http://bit.ly/3jAvlb8.

See a review of *Words In Common* by Canberra poet Michael Byrne at http://bit.ly/2MRlv8y.

**A SELECTION OF MY MATHEMATICAL PAPERS:**

27 CAMPBELL, G. B. *An interview with Rodney James Baxter*, Aust. Math. Soc. Gazette, Volume 47, No1, pp24-32, March 2020. (https://austms.org.au/wp-content/uploads/2020/07/471Web.pdf)

26. CAMPBELL, G. B. Some n-space q-binomial theorem extensions and similar identities, arXiv:1906.07526v1 [math.NT], Jun 2019. (https://arxiv.org/abs/1906.07526)

25 CAMPBELL, G. B. and ZUJEV, A. The series that Ramanujan misunderstood, arXiv:1610.03693v1 [math.NT], Oct 2016. (https://arxiv.org/abs/1610.03693v1)

24 CAMPBELL, G. B. and ZUJEV, A. On integer solutions to x^{5} - (x+1)^{5} - (x+2)^{5} + (x+3)^{5} = 5^{m} + 5^{n}, arXiv:1603.00080v1 [math.NT], Feb 2016. (https://arxiv.org/abs/1603.00080v1)

23 CAMPBELL, G. B. and ZUJEV, A. Some equations with features of digit reversal and powers, arXiv:1602.06320v1 [math.NT], Feb 2016. (https://arxiv.org/abs/1602.06320v1)

22 CAMPBELL, G. B. and ZUJEV, A. Gaussian integer solutions for the fifth power taxicab number problem, arXiv:1511.07424v1 [math.NT], Nov 2015. (https://arxiv.org/abs/1511.07424v1)

21 CAMPBELL, G. B. and ZUJEV, A. Variations on Ramanujan's nested radicals, arXiv:1511.06865v1 [math.NT], Nov 2015. (https://arxiv.org/abs/1511.06865v1)

20 CAMPBELL, G. B. and ZUJEV, A. A diophantine sum with factorials, arXiv:1510.03056v2 [math.NT], Oct 2015. (https://arxiv.org/abs/1510.03056v2)

19 CAMPBELL, G. B. The q-Dixon sum Dirichlet series analogue, arXiv:1302.2664v1, Feb 2013. (https://arxiv.org/abs/1302.2664v1)

18 CAMPBELL, G. B. Ramanujan and Eckford Cohen totients from Visible Point Identities, arXiv:1212.2818v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2818v1)

17 CAMPBELL, G. B. D-analogues of q-shifted factorial and the q-Kummer sum, arXiv:1212.2248v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2248v1)

16 CAMPBELL, G. B. Polylogarithm approaches to Riemann Zeta function zeroes, arXiv:1212.2246v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2246v1)

15 CAMPBELL, G. B. Dirichlet series analogues of q-shifted factorial and the q-Kummer sum, Research paper 2003-6, Department of Mathematics, LaTrobe University, 2003.

14 CAMPBELL, G. B. An Euler Product transform applied to q series, Ramanujan J (2006) 12:267-293. (https://doi.org/10.1007/s11139-006-0078-y)

13 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.

12 CAMPBELL, G. B. Combinatorial Identities in Number Theory related to q-series and Arithmetical functions, Bull. Austral. Math. Soc., Vol. 58, (1998) pp345-347.

11 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.

10 CAMPBELL, G. B. Visible point vector summations from hypercube and hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 21, No 4, 741-748, 1998. (https://www.researchgate.net/publication/26536267_Visible_point_vector_summations_from_hypercube_and_hyperpyramid_lattices)

9 CAMPBELL, G. B. Infinite products over hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 23, No 4, 2000, 271-277. (http://downloads.hindawi.com/journals/ijmms/2000/108918.pdf)

8 CAMPBELL, G. B. A closer look at some new identities, Internat. J. Math. & Math. Sci., Vol 21, No 3, 1998, pp581-586. (https://www.researchgate.net/publication/26536244_A_closer_look_at_some_new_identities)

7 CAMPBELL, G. B. Infinite products over visible lattice points, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 637-654. (http://downloads.hindawi.com/journals/ijmms/1994/705467.pdf)

6 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.

5 CAMPBELL, G. B. Formulae with functions exhibiting self-similarity, Research Paper preprint series, Centre for Mathematics and its Applications, The Australian National University, 1993.

4 CAMPBELL, G. B. A generalised formula of Hardy, Int. J. Math. Math. Sci., Vol 17, No 2, 1994, 369-378.

3 CAMPBELL, G. B. Dirichlet summations and products over primes, Internat. J. Math. & Math. Sci., Vol 16, No 2, 1993, 359-372.

2 CAMPBELL, G. B. Multiplicative functions over Riemann zeta function products, J. Ramanujan Soc. 7 No. 1, 1992, 52-63.

1 CAMPBELL, G. B. Generalization of a formula of Hardy, La Trobe University preprints no 79-5, 1979 (written whilst a young student.)