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


Math 547/702I, Algebraic Structures II (Honors), Spring 2015.

Math 374, Discrete Structures, Spring 2015.

Previously, in reverse chronological order: (with complete notes in some cases!)

Math 546 (Alg I)/a>, 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: Daniel Kamenetsky.

Curriculum vitae.

My talk at the Joint Meetings: 1 + 2 + 3 + 4 + ...

Dartmouth: 1 + 2 + 3 + 4 + ...

Seminars at the University of South Carolina:

Number Theory Seminar, Algebraic Geometry Seminar, and Commutative Algebra Seminar, Spring 2015

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, to appear.

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, accepted for publication.

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.

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.

17. Identities for field extensions generalizing the Ohno-Nakagawa relations, with Henri Cohen and Simon Rubinstein-Salzedo.

Supplementary file:

18. The number of ramified primes in number fields of small degree, with Robert Lemke Oliver.

Several more in preparation.

Upcoming/Recent Professional Activities:

, Winthrop University, Rock Hill, SC, March 28-29.

SQuaRE on Secondary Terms in the Davenport-Heilbronn Theorems, American Institute of Mathematics, San Jose, CA, May 11-15, 2015.

Kobe University, May 16-June 8, 2015.

Joetsu University of Education, Joetsu, Japan, (??)

Elementary, Analytic, and Algorithmic Number Theory (Pomerance 70), June 9-11, University of Georgia.

Wade Into Research (Wake Forest/Davidson REU), ???

Finite Fields and their Applications (maybe), July 13-17, 2015.

Silvermania (maybe), August 11-15, 2015.