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

For 2018-19 I am on sabbatical at Tufts University, working with Robert Lemke Oliver, and supported by a Simons Fellowship.

Current Teaching (Spring 2018)

SCHC 212, The Mathematics of Game Shows.

Math 702, Graduate Algebra II.


Math 374 (Discrete for CS). Math 701 (Alg I), 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).

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

Current Graduate Students: Harsh Mehta.

Curriculum vitae.

Seminars at the University of South Carolina:

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.

Computer code:,,,,,

21. On D_l-extensions of odd prime degree l, with Henri Cohen.

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.

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

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.

Several more in preparation.

Upcoming/Recent Professional Activities:

Stony Brook University, January 25, 2018.

South Carolina High School Math Contest, USC, February 3, 2018.

Southeast Regional Meeting on Numbers, March 10-11, 2018.

Automorphic Forms Workshop, Tufts University, March 19-22, 2018.

College of Charleston, April 6, 2018.

Purdue University, April 10, 2018.

Emory University, April 24, 2018.

Additive Combinatorics from a Geometric Viewpoint (organizer, with Czabarka and Székely), University of South Carolina, May 21-25, 2018. Apply here!

Modular Forms and Related Topics, Beirut, May 28-31, 2018.

Canadian Number Theory Association (to be confirmed), Quebec City, July 9-13, 2018.

Arithmetic Geometry, Number Theory, and Computation, MIT, August 20-24, 2018.

Simons Foundation MPS Annual Meeting, New York, October 18-19, 2018.

Palmetto Number Theory Series, December 2018 (?).

Arizona Winter School on Topology and Arithmetic (?), March 2-6, 2019.

Journees Arithmetiques (?), Istanbul, July 1-5, 2019.