Geoffrey Campbell

Geoffrey Campbell
Long Term Campus Visitor
PhD (ANU), GDip Internet and Web Comp (RMIT University)

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Research interests

In 2022-23 I have written eight "Fun with Numbers" brief articles for the Gazette of the Australian Mathematical Society. These are small articles on simple number theoretic topics.

My main mathematics of 2022-2023 was the book project described below. My 2019 paper's approach to vector partition theory (See was the starting place for the work.

It's a monograph entitled Partitions, Visible Point Vectors and Ramanujan Functions. It covers classical Integer Partitions, Rogers-Ramanujan Partitions in Statistical Mechanics Solved Models, and includes Vector Partitions ideas and the Visible Point Vector identities I found some years ago. It has chapters on Plane Partitions, Asymptotic Partition Formulas, Partition Congruences, Ramanujan Continued Fractions, Polylogarithms, Parametric Euler sum identities, Higher Dimensional Weighted Partition Identities. The final draft is submitted and in editorial flux until planned publication in 2024.

My table of contents, as at November 2023, is:

Partitions, Visible Point Vectors and Ramanujan Functions

   1. The author's vision and motivations
   2. Plans for the book and its academic context

Section I Background and History

Chapter 1. Historical background - The range of literature
   1. Discovering the Ramanujan miracle
   2. Three notable George E Andrews texts on Partitions
      2.1. The Theory of Partitions (1976) by G. E. Andrews
      2.2. Integer Partitions (2004) by G. E. Andrews with K. Eriksson
      2.3. G. E. Andrews 80 Years of Combinatory Analysis (2021)
   3. Basic Hypergeometric Series
   4. Lattice Sums in Chemistry molecular structures
   5. Polylogarithms and computational research related results
   7. Partition theory in Statistical Mechanics and Theoretical Physics

Chapter 2. A brief history timeline for partitions
   1. Leonhard Euler - 18th century
   2. Gauss, Cauchy and Heine - 19th century
   3. Rogers and Ramanujan - 1890s to 1920s
   4. Major Percy MacMahon - 1896 to circa 1920
   5. Hardy and Ramanujan - circa 1918 to the 1940s
   6. Andrews - early 1960s to circa 2020
   7. Rodney Baxter - 1980 to 2023 and beyond

Section II Integer Partition Theory

Chapter 3. Integer partitions and their generating functions
   1. Preview
   2. Euler’s approach to partitions
   3. Euler’s partition identity
   4. Euler’s pentagonal number theorem
   5. The q-binomial Theorem
   6. The Jacobi Triple Product
   7. Two Identities from Gauss
   8. The Watson Quintuple Product
   9. The Heine fundamental transformation
   10. The q-analogues of Gauss’s and Kummer’s Theorems
   11. Euler-Pair theorem
   12. Rogers-Ramanujan identities

Chapter 4. Continued Fractions for Partition Generating Functions
   1. Euler’s Continued Fraction
   2. Euler’s continued fraction applied to partitions
   3. Rogers-Ramanujan Continued Fractions for partition functions
   4. Ramanujan’s three parameter continued fraction
   5. The Ramanujan Machine and computer assisted research
Chapter 5. Congruence properties of partitions
   1. Ramanujan’s partition congruences
   2. 1944, Freeman Dyson and his Hypothesized Crank
   3. 1988, George Andrews and Frank Garvan find Dyson's Crank
   4. Further congruence identities for partitions
   5. Rödseth–Gupta theorem on binary partitions
   6. The 2D and 3D binary partitions congruence theory
   7. Congruences arising from Exponential Generating Functions

Chapter 6. Ferrers Graphs and Ferrers Boards
   1. Ferrers graphs and conjugate graphs
   2. Ferrers Boards
   3. Bijection proofs using Ferrers graphs
   4. Franklin’s near-Bijection proof of Euler’s Pentagonal Number Theorem

Chapter 7. Durfee Squares
   1. What is a Durfee Square?
   2. Durfee Squares applied to Generating Functions
   3. Successive Durfee Squares
   4. Bijection proofs of Rogers-Ramanujan type identities

Chapter 8. Gaussian polynomials
   1. Definition and particular cases
   2. Combinatorial interpretations
      2.1. Inversions
      2.1. Balls into bins
      2.2. Reflection
   3. Analogs of Pascal’s identity
   4. q-binomial theorem analogue
   5. Gaussian polynomials in the Theory of Partitions

Chapter 9. Plane Partitions from MacMahon to Andrews
   1. Definitions of Plane Partitions
   2. Lozenge tilings of a hexagon.
   3. Generating functions for Plane Partitions
   4. The Ten Symmetry Classes of Plane Partitions
     4.1. Class 1: Unrestricted Plane Partitions
     4.2. Class 2: Symmetric Plane Partitions
     4.3. Class 3: Cyclically Symmetric Plane Partitions
     4.4. Class 4: Totally Symmetric Plane Partitions
     4.5. Class 5: Self-Complementary Plane Partitions
     4.6. Class 6: Transpose-Complementary Plane Partitions
     4.7. Class 7: Symmetric Self-Complementary Plane Partitions
     4.8. Class 8: Cyclically Symmetric Transpose-Complementary Plane Partitions
     4.9. Class 9: Cyclically Symmetric Self-Complementary Plane Partitions
     4.10. Class 10: Totally Symmetric Self-Complementary Plane Partitions
   5. The State of Play circa 2023 and Open Questions
   6. The Gog and Magog Trapezoids and Alternating Sign Matrices
   7. Inferences from Plane Partitions that may apply to Vector Partitions

Chapter 10. Asymptotics for partition functions
   1. Ramanujan's early partition asymptotic conjecture
   2. The Circle Method of Hardy and Ramanujan
   3. The Rademacher exact formula
   4. The Theorem of Meinardus
   5. A polynomial analogue of Meinardus’ Theorem
   6. The Convolution Method for products of two series
   7. Bruinier and Ono’s exact formula for p(n)

Chapter 11. The partition function in Statistical Mechanics
   1. Baxter’s Hard Hexagon Model solved exactly by Rogers-Ramanujan identities
   2. Hard Hexagon Model Regime II identities from Baxter
   3. Outline of proofs of Baxter’s Regime II Conjectures
   4. Transforms between Baxter’s Regime II and Regime III.
   5. The partition function of the Hard Hexagon Model
      5.1. Solution
   6. Rogers-Ramanujan shift from Mathematics to Physics
   7. Partition generating functions in Physics are polynomial generalizations
   8. The Elliptic q-Gamma Function

Section III Vector Partition Theory

Chapter 12. Vector partitions and their generating function identities
   1. Defining Vector Partitions
   2. Vector or Multipartite Partitions from Andrews
   3. Defining Vector Grids and resulting Partition Grids
   4. Partitions into exactly two parts, and exactly three parts
   5. First 3D hyperquadrant tableau reduced to a 2D tableau
   6. Vector Partitions whose parts are on lines in nD
      6.1 Any integer partition theorem applies to lines in n-space
      6.2 Weighted integer partitions with function of partition function, 
   7. 2D and 3D Upper Radial Regions of vectors for partition identities
   8. Defining 2D Upper Visible Point Vectors in origin-radial regions
   9. Examples of 2D VPV finite generating functions
      9.1.  2D Distinct Upper VPV Coefficients - Order 2
      9.2.  2D Distinct Upper VPV Coefficients - Order 3
      9.3.  2D Distinct Upper VPV Coefficients - Order 4
      9.4.  2D Distinct Upper VPV Coefficients - Order 5
      9.5.  2D weighted Upper VPV Coefficients - Order 5
   10. Defining radial from origin region 2D Upper All Vectors aggregates
   11. Examples of 2D Upper All Vectors finite generating functions
      11.1.   2D Distinct Upper All Vectors Coefficients - Order 2
      11.2.  2D Distinct Upper All Vectors Coefficients - Order 3
      11.3.  2D Distinct Upper All Vectors Coefficients - Order 4
      11.4.  2D Unrestricted Upper All Vectors Partitions - Order 4
      11.5.  2D Distinct Upper All Vectors Coefficients - Order 5

Chapter 13. Integer Partitions to Vector Partitions
   1. Use of the q-binomial theorem
   2. A higher dimensional kind of q-binomial theorem
   3. Proving the n-space version of q-binomial theorem.
   4. Finite Product 2D, 3D and nD cases of q-binomial extensions.
      4.1. 2D, 3D and nD unrestricted partitions on a cluster of vectors 
      4.2. 2D, 3D and nD distinct partitions on a cluster of vectors
   5. Partitions in 2D Triangle, 3D Pyramid or nD Hyperpyramid Lattices

Chapter 14. Weighted Vector Partitions as hybrid n-space variations
   1. Examples of hybrid variations for identities generating weighted vector
   2. Proof of the n-space hybrid q-binomial theorem.
   3. Exercises

Chapter 15. Functional Equations for n-space Vector Partitions
   1. An n-space q-binomial functional equation.
   2. An n-space binary vector partitions functional equation.
   3. Exercises

Chapter 16. Binary Partitions and their Vector Generalizations
   1. Elementary ideas for binary Partitions
   2. A few finite and infinite products for binary partitions
   3. 2D version of every integer is a unique sum of distinct binary powers
   4. The 2D binary, n-ary and 10-ary formulas
   5. Some binary integer partition preliminary results
   6. Some easy 2D binary partition transform generating functions
   7. A binary partition 2-space simple identity
   8. First quadrant 2D binary partitions
   9. First quadrant upper diagonal 2D binary partitions
   10. First hyperquadrant 3D binary partitions

Chapter 17. n-ary Partitions and their Vector Generalizations
   1. Integer n-ary partitions
   2. A Base 10 or 10-ary set of cases
   3. The n-ary integer partition set of cases
   4. The binary vector partition set of cases

Chapter 18. Some Binary and n-ary Partition Analytic Formulas
   1. History of the binary partition oscillating series
   2. Some lacunary series and products for n-ary partitions
   3. Binary and n-ary versions of Products for Distinct Partitions

Section IV Visible Point Vector Partition Theory

Chapter 19. Features of the Visible Lattice Points
   1. Visible Points and the gcd function
   2. Facts about Visible Points
   3. The Farey fractions relation to the 2D visible points
      3.1 Farey Sunburst
   4. Orchard Visibility and the 2D Visible Lattice Points
      4.1 Polya’s Orchard Visibility Problem
      4.2 Euclid’s Orchard
      4.3 Opaque Forests

Chapter 20. Visible Point Vector Identities in the first Hyperquadrant
   1. Introducing the VPV identities
   2. Deriving the 2D first quadrant VPV identity.
   3. Deriving the n-dimensional first hyperquadrant VPV identity.
   4. Hyperquadrant lattices and their hyperdiagonal line functions
   5. Some hyperdiagonal line generating functions
   6. Diversionary note on the abc Conjecture
   7. Applying a 3D VPV identity to the abc Conjecture
   8. Hyperdiagonal line generating functions for different nD slopes

Chapter 21. Visible Point Vector Identities in Hyperpyramid lattices
   1. VPV identities in square hyperpyramid regions
   2. Deriving 2D VPV identities in extended triangle regions
   3. Deriving 3D VPV identities in square pyramid regions
   4. VPV identities in nD first hyperquadrant hyperpyramid regions
   5. 2D VPV identities for a z-axis symmetric extended triangle lattice
   6. 3D VPV identities for a right square pyramid lattice
   7. VPV identities in nD right-square hyperpyramid regions
   8. Envoi: Research Exercises

Chapter 22. Polylogarithms, and Parametric Euler Sum identities
   1. Purpose of this chapter
   2. Early history of the Dilogarithm
   3. The Trilogarithm function
      3.1. 2D Hyperquadrant VPV Cases
      3.2. 3D Hyperquadrant VPV Cases
      3.3. 4D Hyperquadrant VPV Cases
      3.4. 5D Hyperquadrant VPV Cases
      3.5. More Trilogarithm equations
   4. The Polylogarithm function and Jonathan Borwein
   5. Mordell-Tornheim-Witten ensembles
   6. Finite Euler Sums
      6.1. The 2D square hyperpyramid VPV identity
      6.2. The 3D square hyperpyramid VPV identity
   7. Parametric Euler Sum Identities

Chapter 23. Visible Point Vector identities from particular Euler sum values
   1. Polylogarithms near trivial zeroes of the Riemann zeta function
   2. Identities near non-trivial zeroes of the Riemann zeta function
   3. Non-trivial zeroes of zeta function Re and Im parts

Chapter 24. Visible Point Vector Identities in Skewed Hyperpyramid lattices
   1. Visible Point Vector identities in asymmetric hyperpyramid regions
      1.1. A more general hyperpyramid theorem.

Chapter 25. Harmonic Sums applied to VPV Identities
   1. The Sofo papers on Harmonic Sums
   2. Creating Harmonic nD VPV Identities
   3. Harmonic sums equal to Riemann Zeta combinations
   4. Harmonic Sums and Polylogarithm Integrals for use in VPV Identities

Chapter 26. The Ramanujan trigonometric function and visible point identities
   1. Extending the Ramanujan trigonometric function
   2. Dirichlet series generating functions
   3. Proof of Theorem 26.1
   4. A new Jordan Totient generating function, and some related results
   5. Further multidimensional formulas
   6. Application of Jacobi theta series to the generalized summations

Chapter 27. Other non-weighted n-space Vector Partition Theorems
   1. Simple binary partition 2-space identities
   2. 2D partitions into distinct binary parts on two straight lines
   3. Simple ternary partition 2-space identities

Chapter 28. VPV Identity cases related to some exponential relations
      1. Rational solutions of xy = yx
      2. Solutions to xyyx = vwwv in rationals and integers
      3. VPV Identity transforms using xy = yx.
      4. VPV Identity transforms using xyyx = vwwv.

Section V Models, Interpretations and some Useful Tools

Chapter 29. 2D and 3D Stepping Stones, Forests, Orchards and Light Diffusions
   1. Models for our Vector Partition Theories and Analyses
   2. 2D Stepping Stone Jumps to Visible Points
   3. Stepping Stone weight values for jumps
   4. Heuristic concepts for the known results
   5. Orchard Visibility and the 2D Visible Lattice Points
      5.1. Polya’s Orchard Visibility Problem
      5.2. Euclid’s Orchard
      5.3. Opaque Forests
   6. Light Diffusion Models
      6.1. Light Diffusion VPV n-space Identities

Chapter 30. Euler Products over Primes and new VPV Formulas
   1. Euler products over the primes
   2. Further Euler products from VPV cases
   3. Simple but new Products Over Primes

Chapter 31. Determinants, Bell Polynomial Expansions for Vector Partitions
   1. Some reference cases of the determinants in this book
   2. Calculus of Determinant Evaluation

Chapter 32. Glossary


Other projects:

Dirichlet series analogues of q-series, where arithmetical function identities encode quasicrystals. I give new analogue summations for classical q-series and hypergeometric series summations in terms of Riemann Zeta functions and Jordan Totient functions. This is a further monograph for me to complete.

So I have research interests 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.

I am Manager of the almost 40,000 member LinkedIn Number Theory Group located at

I am an Administrator for the Facebook Group Classical Mathematics at presently with about 20,000 members.


Room 2.73, Hanna Neumann Building 145


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

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
See a review of Words In Common by Canberra poet Michael Byrne at


41            CAMPBELL, G. B. Vector Partitions, Visible Points and Ramanujan Functions, Chapman & Hall/CRC, to appear, 2024. 

40            CAMPBELL, G. B. Fun with numbers: Ternary or Base Three Identities, Aust. Math. Soc. Gazette, Volume 50, No5, November 2023. (

39            CAMPBELL, G. B. Visible Point Vector Partition Identities for Hyperpyramid Lattices arXiv:2309.16094 [math.CO]. ( September 2023.

38            CAMPBELL, G. B. Fun with numbers: Multigrade Sums with Carl Sagan and Pell Equations, Aust. Math. Soc. Gazette, Volume 50, No4, September 2023. (

37            CAMPBELL, G. B. Fun with numbers: Any rational is a sum of four 4th and four 5th powers Aust. Math. Soc. Gazette, Volume 50, No3, pp.117-119, July 2023. (

36            CAMPBELL, G. B. Visible Point Partition Identities for Polylogarithms, and Parametric Euler Sums,  arXiv:2306.02241 [math.CO]. ( June 2023.

35            CAMPBELL, G. B. Fun with numbers: Revisiting an Eulerian Problem Aust. Math. Soc. Gazette, Volume 50, No2, pp.10-12, May 2023. (

34            CAMPBELL, G. B. Fun with numbers: A base 7 identity Aust. Math. Soc. Gazette, Volume 50, No1, pp.13-14, March 2023. (

33            CAMPBELL, G. B. Vector Partition Identities for 2D, 3D and nD LatticesarXiv:2302.01091v1 [math.CO], Feb 2023. (

32            CAMPBELL, G. B. Continued Fractions for partition generating functionsarXiv:2301.12945v1 [math.CO], Jan 2023. (

31            CAMPBELL, G. B. Fun with numbers: Rational solutions to xyyx = vwwv Aust. Math. Soc. Gazette, Volume 49, No5, pp210-211, November 2022. (

30            CAMPBELL, G. B. Fun with numbers: Identities containing a certain algebraic form Aust. Math. Soc. Gazette, Volume 49, No4, pp162-163, September 2022. (

29            CAMPBELL, G. B. Fun with numbers: Consecutive 6th powers and base 6 numbers,  Aust. Math. Soc. Gazette, Volume 49, No3, pp108-109, July 2022. (

28            CAMPBELL, G. B. Fun with numbers: Ramanujan 6-10-8 identity,  Aust. Math. Soc. Gazette, Volume 49, No2, pp71-72, May 2022. (

27            CAMPBELL, G. B. An interview with Rodney James Baxter, Aust. Math. Soc. Gazette, Volume 47, No1, pp24-32, March 2020. (

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

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

24            CAMPBELL, G. B. and ZUJEV, A. On integer solutions to x5 - (x+1)5 - (x+2)5 + (x+3)5 = 5m + 5n, arXiv:1603.00080v1 [math.NT], Feb 2016. (

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

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

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

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

19            CAMPBELL, G. B. The q-Dixon sum Dirichlet series analogue, arXiv:1302.2664v1, Feb 2013. (

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

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

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

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. (

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. (

9              CAMPBELL, G. B. Infinite products over hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 23, No 4, 2000, 271-277. (

8              CAMPBELL, G. B. A closer look at some new identities, Internat. J. Math. & Math. Sci., Vol 21, No 3, 1998, pp581-586. (

7              CAMPBELL, G. B. Infinite products over visible lattice points, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 637-654. (

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.)

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