Frank Thorne
University of South Carolina |

Math 788, The Geometry of Numbers, Spring 2014.

Math 580, Elementary Number Theory, Fall 2013.

Math 142, Calculus II, Fall 2013.

Math 788p, Topics in Algebraic Number Theory, Spring 2013.

Reading seminars: Complex analysis, Mondays at 10:00 am; Algebraic geometry, Wednesdays at 9:30 am.

Math 531, Foundations of Geometry, Fall 2012.

Math 141, Calculus I, Fall 2012.

Math 574, Discrete Math, Spring 2012.

Math 141, Calculus I, Fall 2011.

Math 782, Analytic Number Theory, Fall 2011.

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

- Classical analytic number theory. My first task in graduate school was to read all of Davenport. I am interested in sieve methods, L-functions, and related topics, and their classical applications -- as well as applications beyond their traditional settings.
- The distribution of number fields, class group torsion, and related topics. I am especially
interested in these from an analytic point of view involving
*Shintani zeta functions*associated to*prehomogeneous vector spaces*. I am also interested in other perspectives on these questions: class field theory and Kummer theory; the Scholz reflection principle; algebro-geometric perspectives on these topics; and other related perspectives.

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

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

Number Theory Seminar, Spring 2013

South Carolina Algebraic Geometry, Commutative Algebra, Number Theory, Fall 2013

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*, to appear.

8.
Shintani's zeta function is not a
finite sum of Euler products.

*
Proceedings of the American Mathematical Society*, to appear.

9.
Orbital L-functions for the space of binary cubic forms, with Takashi Taniguchi.

*
Canadian Journal of Mathematics*, to appear.

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,* to appear.

12.
Book review of *Opera de Cribro* and *An introduction to sieve methods and their applications*.

*
Bulletin of the American Mathematical Society*, to appear.

13.
On the existence of large degree Galois representations for fields of small discriminant,
with Jeremy Rouse.

*
Submitted.*

14.
Dirichlet series associated to cubic fields with given quadratic resolvent,with Henri Cohen.

*
Michigan Math Journal,* to appear.

15.
Dirichlet series associated to quartic fields with given resolvent, with Henri Cohen.

*
Submitted.*

16.
Zeros of L-functions outside the critical strip, with Andrew Booker.

*
Submitted.*

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

*
Submitted.*

Several more in preparation.

Palmetto Number Theory Series, Clemson, December 7-8, 2013.

Pro p-groups and arithmetic, Besançon, December 16-20, 2013.

Automorphic Forms and Arithmetic, Gottingen, February 10-14, 2014.

University of Bonn, March 4-14, 2014.

Zürich Number Theory Days, March 7-8, 2014.

Arizona Winter School, Tucson, March 15-19, 2014.

Southeast Regional Meeting on Numbers, Wofford College, April 19-20, 2014.

Ursinus College, early May.

Bateman and Halberstam, University of Illinois, June 5-7, 2014.

Counting Arithmetic Objects, Montreal, June 23-July 4, 2014.

Richard Oh's thesis defense, July 8, 2014.

Emerging Leaders and Evolving Frontiers in Analytic Number Theory, Bonn, July 14-18, 2014.

Gangnam Style, August 13-21, 2014.

Emory University, August 27, 2014.

Workshop on Counting Arithmetic Objects, Montreal, November 10-14, 2014.

Analysis, Spectra, and Number Theory, Princeton, December 15-19, 2014.