Frank Thorne

University of South Carolina
Department of Mathematics
317-O LeConte College
1523 Greene Street
Columbia, SC 29201
thorne [at] math [dot] sc [dot] edu

Current Teaching (Fall 2021

Math 532, Modern Geometry.

Math 701, Graduate Algebra I.


Math 550 (Vector), Math 788 (Ell. curves), Math 142 (Calc II), Math 782 (The Distribution of the Primes), Math 374 (Discrete for CS), Math 701-2 (Alg I/II), Math 141 (Calc I), Math 735 (Lie Groups), Math 788 (Ell. curves), Math 141 (Calc I), Math 544 (Lin Alg), Math 547/702I (Alg II), Math 374 (Discrete for CS), Math 546 (Alg I), Math 142H, (Calc II), Math 788G (GON), Math 580 (Intro NT), Math 142 (Calc II), Math 788P (Alg NT), Math 531 (Geometry), Math 141 (Calc I), Math 574 (Discrete), Math 141 (Calc I), Math 782 (Anal NT), (2020-21 links to be added).

The Mathematics of Game Shows (with course notes -- feel free to use!)

I work in number theory. Some of my particular interests:

Current Graduate Students: Demmas Salim.

Curriculum vitae.

Seminars at the University of South Carolina:

(links out of date)

Algebraic Geometry, Arithmetic Geometry, and Commutative Algebra, on Friday afternoons.

Number Theory, on Thursday mornings.

The colloquium.

Graduate students are also encouraged to attend the graduate colloquium.

See here for a list of other seminars run at USC.

Publications and Preprints:

1. Bounded gaps between products of primes with applications to elliptic curves and ideal class groups.
International Mathematics Research Notices (2008), 41 pp.

2. Irregularities in the distributions of primes in function fields.
Journal of Number Theory 128 (2008), 1784-1794.

3. Bubbles of congruent primes.
Mathematical Proceedings of the Cambridge Philosophical Society 157 (2014), no. 3, 443–456.

4. An uncertainty principle for function fields.
Journal of Number Theory 131 (2011), 1363-1389.

5. Maier matrices beyond Z.
Proceedings of the Integers Conference 2007.

6. Analytic properties of Shintani zeta functions.
Proceedings of the RIMS Symposium on automorphic forms, automorphic representations, and related topics (Kyoto, 2010).

7. The secondary term in the counting function for cubic fields, with Takashi Taniguchi.
Duke Mathematical Journal 162 (2013), no. 13, 2451-2508.

8. Shintani's zeta function is not a finite sum of Euler products.
Proceedings of the American Mathematical Society, 142 (2014), no. 6, 1943-195 2.

9. Orbital L-functions for the space of binary cubic forms, with Takashi Taniguchi.
Canadian Journal of Mathematics 65 (2013), no. 6, 1320-1383.

10. Four perspectives on secondary terms in the Davenport-Heilbronn theorems.
Integers Volume 12B, Proceedings of the Integers Conference 2011.

11. An error estimate for counting S_3-sextic number fields, with Takashi Taniguchi.
International Journal of Number Theory 10 (2014), no. 4, 935-948.

Supplementary files:,, and an old note with some supplementary numerical data.

More supplementary files:,, and

12. Book review of Opera de Cribro and An introduction to sieve methods and their applications.
Bulletin of the American Mathematical Society 50 (2013), no. 2, 359-366.

13. On the existence of large degree Galois representations for fields of small discriminant, with Jeremy Rouse.
Pacific Journal of Mathematics 271 (2014), no. 1, 243-256.

14. Dirichlet series associated to cubic fields with given quadratic resolvent,with Henri Cohen.
Michigan Math Journal 63 (2014), no. 2, 253-273.

See below [11] for source code; also: a short list of cubic fields generated by Karim Belabas's program.

15. Dirichlet series associated to quartic fields with given resolvent, with Henri Cohen.
Research in Number Theory 2 (2016), Art. 29, 40 pp.

Appendix: Complete proof of Theorem 5.1, whose proof was only sketched in the paper.

16. Zeros of L-functions outside the critical strip, with Andrew Booker.
Algebra and Number Theory 8 (2014), no. 9, 2027-2042.

17. Identities for field extensions generalizing the Ohno-Nakagawa relations, with Henri Cohen and Simon Rubinstein-Salzedo.
Compositio Mathematica 151 (2015), no. 11, 2059-2075.

Supplementary file:

18. The number of ramified primes in number fields of small degree, with Robert Lemke Oliver.
Proceedings of the American Mathematical Society 145 (2017), no. 8, 3201–3210.

19. Analytic number theory and zeta function methods.
Submitted. Solicited for inclusion in the Proceedings of the 2014 Summer School on Counting Arithmetic Objects, Montreal.

20. Orbital exponential sums for prehomogeneous vector spaces, with Takashi Taniguchi.
American Journal of Mathematics.

Computer code:,,,,,

21. On D_l-extensions of odd prime degree l, with Henri Cohen.
Journal of the London Math Society, to appear.

22. Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves.
  With Manjul Bhargava, Arul Shankar, Takashi Taniguchi, Jacob Tsimerman, and Yongqiang Zhao.
Journal of the American Mathematical Society, to appear.

23. Levels of distribution for sieve problems in prehomogeneous vector spaces, with Takashi Taniguchi.
Mathematische Annalen, to appear.

24. The distribution of G-Weyl CM fields and the Colmez conjecture, with Adrian Barquero-Sanchez and Riad Masri.

25. Uniform bounds for lattice point counting and partial sums of zeta functions, with David Lowry-Duda and Takashi Taniguchi.

26. Rank growth of elliptic curves in nonabelian extensions, with Robert Lemke Oliver.
International Mathematics Research Notices, to appear.

27. Upper bounds on polynomials with small Galois group, with Robert Lemke Oliver.
  Mathematika, to appear.

28. Improved lower bounds for the number of fields with alternating Galois group, with Aaron Landesman and Robert Lemke Oliver.

29. Malle's Conjecture for G×A, with G=S_3,S_4,S_5, with Riad Masri, Wei-Lun Tsai, and Jiuya Wang.

30. Upper bounds on number fields of given degree and bounded discriminant, with Robert J. Lemke Oliver.

31. Counting quintic fields with genus number one, with Kevin J. McGown and Amanda Tucker.

32. Asymptotic identities for additive convolutions of sums of divisors, with Robert J. Lemke Oliver and Sunrose Shrestha.
To be submitted shortly.

Several more in preparation.

Upcoming/Recent Professional Activities:

International Conference on Number Theory and Discrete Mathematics, Kochi, India, December 11-14, 2020.