# Geoffrey Campbell

## Contacts

### Research interests

In statistical mechanics I am interested in exact solutions of models for phase transitions. In a more mathematical direction, I study lattice sums and the asymptotic properties of special functions. I also am very keen on the theory of integer partitions, and have developed this in a vector partition context.

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.

I therefore can say I am interested in Aperiodic Order, Dynamical Systems, Combinatorics, Discrete Geometry, Applied Algebra and Number Theory, Mathematical Biology, and Mathematical Physics, and areas where these regimes seem to arise.

In 2019 I have written 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)

I am also Manager of the 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 six years. A typical simple post is for equations related to (2ⁿ + 1)⁸ - (2ⁿ + i)⁸ - (2ⁿ - i)⁸ + (2ⁿ - 1)⁸; examples of which are 3⁸ - (2 + i)⁸ - (2 - i)⁸ + 1⁸ = 2¹³ - 2¹⁰ + 2⁹ - 2⁶, 5⁸ - (4 + i)⁸ - (4 - i)⁸ + 3⁸ = 2¹⁹ - 2¹⁶ + 2¹¹ - 2⁸, ... , 4097⁸ - (4096 + i)⁸ - (4096 - i)⁸ + 4095⁸ = 2⁷⁹ - 2⁷⁶ + 2³¹ - 2²⁸, etc. While such statements are not 'cutting edge research' nor much more than curiosities, they do sometimes give new results that inform other researches.

### Groups

- Collaborator, Mathematical physics

**A SELECTION OF MY PAPERS:**

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.)