Research

My research areas are number theory and combinatorics. I am especially interested in the areas of partitions, modular and quasi-modular forms, and q-multiple zeta values, as well as applications of these topics to other areas of mathematics. My most recent work and primary focus are on the distribution of partition statistics and the arithmetic, algebraic and number-theoretic properties of q-multiple zeta values. I have linked to the pages of the many people with whom I have collaborated at the end of the page.

Articles in Preparation

1. W. Craig, J. C. Lagarias. Sets of integers with prescribed non-unitary exponent partitions. In preparation.

2. K. Banerjee, K. Bringmann, W. Craig. To be determined.

3. W. Craig, P. Cuthbertson, R. Schneider. To be determined.

Submitted Articles

1. K. Bringmann, C. Cossaboom, W. Craig. Overpartitions with parts separated by parity. Submitted for publication.

2. K. Bringmann, W. Craig, K. Ono. Ramanujan's partition generating functions modulo ℓ. Submitted for publication.

3. K. Bringmann, W. Craig, C. Nazaroglu. Precision asymptotics for partitions featuring false-indefinite theta functions. Submitted for publication.

4. W. Craig. Higher level q-multiple zeta values with applications to quasimodular forms and partitions. Submitted for publication.

5. K. Bringmann, W. Craig, J.-W. van Ittersum, B. V. Pandey. Limiting behaviour and modular completions of MacMahon-like q-series. Submitted for publication.

6. W. Craig, M. L. Dawsey, G.-N. Han. Inequalities and asymptotics for hook numbers in restricted partitions. Submitted for publication.

Accepted Articles

1. W. Craig, R. Li. Distribution of alternating sums of parts in partitions. Accepted for publication in Ramanujan Journal.

2. W. Craig, K. Ono, A. Singh. Distribution of hooks in self-conjugate partitions. Accepted for publication in Discrete Mathematics.

3. K. Bringmann, W. Craig, J. Males. Asymptotics for d-fold partition diamonds and related infinite products. Michigan Mathematical Journal, In Press.

Published Articles

1. W. Bridges, W. Craig, A. Folsom, L. Rolen. Zeros of hook polynomials and related questions. Accepted for publication in SIGMA: Symmetry, Integrability and Geometry. Methods and Applications. (arxiv)

2. W. Craig, W. Raji. The Riemann Hypothesis for period polynomials of cusp forms. Acta Arithmetica 218 (2025), 175-195. (arxiv)

3. W. Bridges, W. Craig. On the distribution of the norm of partitions. The Ramanujan Journal (Special Issue for Proceedings of 100 Years of Mock Theta Functions) 66 (2), 1-5 (2025). (arxiv)

4. K. Bringmann, W. Craig, C. Nazaroglu. On the asymptotic behavior for partitions separated by parity. Electronic Journal of Combinatorics 32, 1 (2025). (arxiv)

5. W. Craig, J.-W. van Ittersum, K. Ono. Integer partitions detect the primes. Proceedings of the National Academy of Sciences, USA 121, 39 (2024), e2409417121. (arxiv) (Quanta) (Nat. Acad. Sci. Cozzarelli Prize) (Scientific American)

6. G. Cesana, W. Craig, J. Males. Asymptotic equidistribution for partition statistics and topological invariants. Journal of Number Theory 256, 373-396 (2024). (arxiv)​​

7. W. Craig. Seaweed algebras and the index statistic for partitions. Journal of Mathematical Analysis and Applications 528, 2 (2023). (arxiv)

8. C. Ballantine, H. Burson, W. Craig, A. Folsom, B. Wen. Hook length biases and general linear partition inequalities. Research in the Mathematical Sciences 10, 41 (2023). (arxiv)

9. J. Balakrishnan, W. Craig, K. Ono, W-L. Tsai. Variants of Lehmer's speculation for newforms. Advances in Mathematics 428, 1 (2023). (arxiv) (Corrigendum)

10. J. Balakrishnan, W. Craig, K. Ono. Variations of Lehmer's conjecture for Ramanujan's tau-function. Journal of Number Theory (JNT Prime and Special Issue on Modular forms and Function Fields) 237, 3-14 (2022). (arxiv)

11. W. Craig. On the distribution of parts into congruence classes in partitions with distinct parts. Research in Number Theory 8, 52 (2022). (arxiv)

12. K. Bringmann, W. Craig, J. Males, K. Ono. Distributions on partitions arising from Hilbert schemes and hook lengths. Forum of Mathematics, Sigma 10, E49 (2022). (arxiv)

13. W. Craig, M. Merca. On Ramanujan-type congruences for multiplicative functions. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 128 (2022). (arxiv)

14. ​W. Craig, P. Linnell. Unique product groups and congruence subgroups. Journal of Algebra and Its Applications 21, 2 (2022). (arxiv)​

15. W. Craig, A. Pun. Distribution properties for t-hooks in partitions. Annals of Combinatorics 25, 677-697 (2021). (arxiv)

16. W. Craig, A. Pun. A note on the higher order Turán inequalities for k-regular partitions. Research in Number Theory 7, 1 (2021). (arxiv)

Ph.D. Thesis

• W. Craig. On the coefficients of q-series and modular forms.

Expository Papers

1. C. Ballantine, H. Burson, W. Craig, A. Folsom, and B. Wen. On the number of hooks of fixed length in odd versus distinct partitions. Seminaire Lotharingien de Combinatoire 89B, Proceedings of the 35th International Conference on Formal Power Series and Algebraic Combinatorics, 12 pp. (2023).​

2. W. Craig, A. Pun. Higher order Turán inequalities for k-regular partitions. Seminaire Lotharingien de Combinatoire 84B, Proceedings of the 32th International Conference on Formal Power Series and Algebraic Combinatorics, 8 pp. (2020).

3. W. Craig. Ketchup science. Math Horizons (April 2018), 16-17.

Collaborators

I have been greatly blessed with the opportunity to work with many wonderful collaborators. I have done my best to list here every person with whom I have coauthored a paper along with links to their personal websites, which I hope I can always keep up to date. I would like to provide a major emphasis on Prof. Ken Ono and Prof. Dr. Kathrin Bringmann, my Ph.D. advisor and postdoctoral advisor respectively, who have been wonderful teachers and collaborators.

1. Jennifer Balakrishnan - Professor, Boston University (Boston, MA, USA)

2. Christina Ballantine - Professor, College of the Holy Cross (Worcester, MA, USA)

3. Alexander Berkovich - Professor, University of Florida (Gainsville, FL, USA)

4. Walter Bridges - Assistant Professor, University of North Texas (Denton, TX, USA)

5. Kathrin Bringmann - Professor, Universität zu Köln (Cologne, Germany)

6. Hannah Burson - Lecturer, University of Minnesota Twin Cities (Minneapolis, MN, USA)

7. Giulia Cesana - Ph.D. student, Universität zu Köln (formerly) (Cologne, Germany)

8. Catherine Cossaboom - Ph.D. student, Oxford University (Oxford, UK)

9. Maddie Dawsey - Associate Professor, University of Texas at Tyler (Tyler, TX, USA)

10. Aritram Dhar - Postdoc, University of Florida (Gainsville, FL, USA)

11. Amanda Folsom - Professor, Amherst College (Amherst, MA, USA)

12. Guo-Niu Han - Professor, Université de Strasbourg (Strasbourg, France)

13. Jan-Willem van Ittersum - Postdoc, Universität zu Köln (Cologne, Germany)

14. Jeffery Lagarias - Professor, University of Michigan (Ann Arbor, MI, USA)

15. Runqiao Li - Ph.D. student, Penn State University (State College, PA, USA)

16. Peter Linnell - Professor, Virginia Tech (R.I.P.) (Blacksburg, VA, USA)

17. Joshua Males - Formerly Postdoc, University of Bristol (formerly) (Bristol, England)

18. Mircea Merca - Professor, Politehnica University of Bucharest (Bucharest, Romania)

19. Caner Nazaroglu - Postdoc, Universität zu Köln (Cologne, Germany)

20. Ken Ono - Professor, University of Virginia (Charlottesville, VA, USA)

21. Badri Vishal Pandey - Postdoc, Universität zu Köln (Cologne, Germany)

22. Anna Pun - Assistant Professor, CUNY-Baruch College (New York City, NY, USA)

23. Wissam Raji - Professor, American University of Beirut (Beirut, Lebanon)

24. Larry Rolen - Assistant Professor, Vanderbilt University (Nashville, TN, USA)

25. Ajit Singh - Postdoc, University of Virginia (Charlottesville, VA, USA)

26. Wei-Lun Tsai - Assistant Professor, University of South Carolina (Columbia, SC, USA)

27. Boya Wen - Assistant Professor, University of Richmond (Richmond, VA, USA)