Frank Thorne
University of South Carolina |

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:

- 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, Algebraic Geometry, and Commutative Algebra Seminar, Spring 2015

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.

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.

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.*

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

*
Submitted.*

Several more in preparation.

Emory University, August 27, 2014.

Palmetto Number Theory Series, South Carolina State, Orangeburg, SC, September 6-7, 2014.

AMS Sectional Meeting, Greensboro, NC, November 8-9, 2014.

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

Palmetto Number Theory Series, University of South Carolina, December 6-7, 2014.

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