# Caroline Turnage-Butterbaugh

Associate Professor of Mathematics and Statistics, Mathematics and Statistics

## Education & Professional History

Wofford College, BA; Wake Forest University, MA; University of Mississippi, PhD

### Previous Appointments

- William W. Elliott Assistant Research Professor, Duke University, 2015-2018
- Postdoctoral Fellow, Mathematical Sciences Research Institute, Spring 2017
- Postdoctoral Research Fellow, North Dakota State University, 2014-2015
- Postdoctoral Undergraduate Research Mentor, Williams College, Summer 2014

At Carleton since 2018.

## Highlights & Recent Activity

- NSF CAREER DMS-2239681 (2023-2029, PI)

## Organizations & Scholarly Affiliations

- American Mathematical Society
- Association for Women in Mathematics
- Mathematical Association of America
- Math Alliance
- National Association of Mathematicians

### Research Interests

My research interests lie in the field of analytic number theory. In particular, I study properties of L-functions (such as the Riemann zeta-function, Dedekind zeta-functions, Artin L-functions, and automorphic L-functions), their zeros, and class groups of number fields.

My work is partially supported by the National Science Foundation under grant NSF CAREER DMS-2239681. It was previously supported by NSF DMS-1902193 (2019-2023, PI) and NSF FRG DMS-1854398 (2019-2023, Senior Scientist Consultant).

## Research Related Travel & Talks

### 2024

**Reverse Site Visit**

October 8, 2024

National Science Foundation

Alexandria, VA**Analytic Number Theory and Arithmetic Statistics – A conference celebrating the mathematical contributions of Chantal David**(invited speaker)

August 26-30, 2024

Centre de recherches mathématiques

Montréal, Québec, Canada

**International Conference on L-functions and Automorphic Forms**(invited speaker)

May 13-16, 2024

Vanderbilt University

Nashville, TN

**Number Theory Seminar**

April 17, 2024

Kansas State University

Manhattan, KS

**Analytic Number Theory**(invited participant and speaker)

March 10-19, 2024

Institut Mittag-Leffler

Djursholm, Sweden

**Pittsburgh Links among Analysis and Number Theory (PLANT)**(invited speaker)

March 5-7, 2023

University of Pittsburgh and Carnegie Mellon University

Pittsburgh, PA

### 2023

**Number Theory Seminar & Department of Mathematics Colloquium**

November 28 & 30, 2023

University of Illinois Urbana-Champaign

Urbana, IL

**Department of Mathematics Colloquium**

November 16, 2023

Vanderbilt University

Nashville, TN

**AWM Career Panel**(invited panelist)

November 10, 2023

Syracuse University (virtual)

**PIMS Network Wide Graduate Courses**(guest lecturer)

November 9, 2023

MATH 740: Moments of L-functions, Instructor: Alia Hamieh (UNBC)

(virtual)

**Joint Athens-Atlanta Number Theory Seminar**

October 24, 2023

Georgia Institute of Technology

Atlanta, GA

**AIM Workshop: Delta Symbols and the Subconvexity Problem**

October 16-20, 2023

American Institute of Mathematics

Pasadena, CA

**AMS Special Session on Analytic Number Theory and Related Fields**

October 7-8, 2023

2023 Fall Central Sectional Meeting, Creighton University

Omaha, NE

**Arithmetic, Algebra, and Algorithms**(invited participant)

April 10-14, 2023

International Centre for Mathematical Sciences

Edinburgh, Scotland

**Algebra/Number Theory Seminar**

March 15, 2023

University of Rochester

Rochester, NY

### 2022

**Number Theory Seminar**

December 7 , 2022

Duke University

Durham, NC

**Number Theory Days**(invited speaker)

September 30 – October 1, 2022

École Polytechnique Fédérale de Lausanne

Lausanne, Switzerland**Focused Research Group (FRG) Graduate Seminar**

September 22, 2022

American Institute of Mathematics (Virtual)

**Duke University**(research visit)

September 2, 2022

Durham, NC

**American Institute of Mathematics**(FRG research visit)

August 8-12, 2022

San Jose, California

**ANTeater ’22**(invited speaker)

August 1-4, 2022

University of California – Irvine

Irvine, California

**PRiME Colloquium**

July 22, 2022

Pomona College (Virtual)

**PIMS CRG Workshop on Moments of L-functions**

July 25-29, 2022

University of Northern British Columbia

Prince George, BC, Canada (Virtual)

**50 Years of Number Theory & Random Matrix Theory Conference**(invited participant)

June 21-24, 2022

Institute for Advanced Study

Princeton, New Jersey

**AWM Research Symposium Special Session: Number Theory**(co-organizer)

June 16-19, 2022

The Institute for Mathematics and its Applications, University of Minnesota

Minneapolis, Minnesota

**47th Annual New York Regional Graduate Mathematics Conference**(plenary speaker)

April 2-3, 2022

Syracuse University

Syracuse, New York

**American Institute of Mathematics**(FRG research visit)

February 3 – March 25, 2022

San Jose, California

(session cancelled)**AMS Special Session: Number Theory at Non-PhD Granting Institutions II**

TBD (was January 7, 2022)

Joint Mathematics Meetings (Conference Postponed)

Seattle, Washington

### 2021

**Algebraic Geometry and Number Theory Seminar**

October 20, 2021

Chalmers University of Technology / University of Gothenburg (Virtual)**Focused Research Group (FRG) Graduate Seminar**

October 4, 2021

American Institute of Mathematics (Virtual)

**American Institute of Mathematics**(FRG research visit)

August 18 – 31, 2021

San Jose, California

**Wofford College Summer Research Seminar**

June 28, 2021

Spartanburg, South Carolina

**Vanderbilt Number Theory Seminar**

May 5, 2021

Vanderbilt University (Virtual)

**Quebec-Vermont Number Theory Seminar**

April 29, 2021

Université de Montréal (Virtual)

**UVA Number Theory Seminar**

April 23, 2021

University of Virginia (Virtual)

**Purdue Analytic Number Theory and Harmonic Analysis seminar (PANTHA)**

March 31, 2021

Purdue University (Virtual)**Focused Research Group (FRG) Seminar**

March 16, 2021

American Institute of Mathematics (Virtual)

**KSU Number Theory Seminar**

March 8, 2021

Kansas State University (Virtual)

**Virtual Brazilian Number Theory Seminar (ViBraNT)**

February 25, 2021

Universidade Federal de Minas Gerais, Brazil (Virtual)

**UM Algebra and Number Theory Seminar**

February 4, 2021

University of Mississippi (Virtual)

### 2020

**Automorphic Forms and Arithmetic**

August 30 – September 5, 2020

Mathematisches Forschungsinstitut Oberwolfach, Germany (Virtual)

**MathFest**(Project NExT Fellow)

July 28 – August 1, 2020

Philadelphia, PA (Virtual)

**VaNTAGe****Seminar**

Tuesday, February 18, 2020

Broadcasting live from Northfield, MN!

**Joint Mathematics Meetings**(invited speaker and Project NExT fellow)

January 15-18, 2020

Denver, CO

### 2019

**Duke University**(research visit)

December 11-20, 2019

Durham, NC

**Analytic Number Theory**

November 3 – 9, 2019

Mathematisches Forschungsinstitut Oberwolfach, Germany

**American Institute of Mathematics**(FRG research visit)

August 27 – September 4, 2020

San Jose, CA

**MathFest**(Project NExT Fellow)

July 31 – August 3, 2019

Cincinnati, Ohio

**NSF-CBMS Conference: L-functions and Multiplicative Number Theory**(invited speaker)

University of Mississippi, May 20-24, 2019

Oxford, Mississippi

**Analytic Aspects of Automorphic Forms**(invited speaker)

Université de Lille, April 3-5, 2019

Lille, France

**Value Distribution of Zeta and L-functions and Related Topics**(invited speaker)

RIKEN Institute March 21 – 27, 2019

Wako, Japan

### 2018

**CMS 2018 Winter Meeting**(invited speaker)

Special Session: Distributions in Analytic Number Theory

Vancouver, Canada

December 7 – 10, 2018

**University of Minnesota – Number Theory Seminar**(invited speaker)

Minneapolis, MN

December 3, 2018

**LMS-CMI Research School: New Trends in Analytic Number Theory**(invited speaker)

Exeter, United Kingdom

August 13 – 17, 2018

slides

**Max Planck Institute for Mathematics**(research visit)

Bonn, Germany

July 1 – 31, 2018

• Number Theory Lunch Seminar (July 18, 2018)

• PLeaSANT Seminar (July 25, 2018)**Number Theory Seminar**

Mathematical Institute, University of Cologne

Cologne, Germany

July 10, 2018**HCM School: L-functions: Open Problems and Current Methods**(course lecturer)

Bonn, Germany

June 25 – 29, 2018**Perspectives on the Riemann Hypothesis**

Bristol, United Kingdom

June 4 – 7, 2018**Institute for Advanced Study**(research visit)

Princeton, New Jersey

May 7 -10, 2018**Number Theory Seminar**

University of New South Wales Sydney

Sydney, Australia

March 13, 2018**Pure Maths Seminar**

University of New South Wales Sydney

Sydney, Australia

March 13, 2018**Department Colloquium**

University of New South Wales Canberra

Canberra, Australia

March 8, 2018**Joint Mathematics Meetings**

San Diego, CA

January 10 – 13, 2018

### 2017

**Department Colloquium**

Oberlin College, Oberlin, OH

November 29, 2017**Algebra, Geometry, and Number Theory Seminar**

Tufts University, Boston, MA

November 16, 2017**Mathematics & Statistics Department Colloquium**

Elon University, Elon, NC

October 18, 2017**WIN4**

Banff International Research Station, Canada

August 14 – 18, 2017**MCA 2017**

Special Session Advances in algebraic and analytic number theory (co-organizer)

Montréal, Canada

July 24 – 28, 2017**SWIM 2017**(course lecturer)

Duke University, Durham, NC

July 20 – 30, 2017**Careers in Academia**

AIM, San Jose, California

June 6 – 9, 2017**re:boot number theory 2017**(co-organizer)

Duke University, Durham, NC

June 1 – 3, 2017**AMS Spring Western Sectional Meeting**

Special Session Analytic number theory and automorphic forms (invited speaker)

Washington State University

April 22 – 23, 2017**Algebraic Geometry and Number Theory Seminar**

Rice University

April 11, 2017**MSRI Analytic Number Theory Semester Program**

Connections for Women (invited speaker)

Program Member (postdoc)

MSRI, Berkeley, CA

January 17 – May 26, 2017**Joint Mathematics Meetings**

AMS Special Session Analytic number theory and arithmetic (invited speaker)

MAA Invited Paper Session L-functions and other animals (co-organizer)

Atlanta, Georgia

January 4 – 7, 2016

### 2016

**Number Theory Seminar**

University of South Carolina

November 17, 2016**Harkness Table Series**

Spartanburg Day School

October 21, 2016**Moments of zeta and correlations of divisor sums**

AIM, San Jose, California

August 29 – September 2, 2016**WADE Into Research**(invited speaker)

Wake Forest University, Winston-Salem, NC

June 24, 2016**SWIM**(invited speaker)

Duke University, Durham, NC

June 21, 2016**Winthrop University REU**(invited speaker)

Winthrop University, Rock Hill, SC

June 17, 2016**Number Theory Seminar**

University of Illinois Urbana-Champaign

April 21, 2016**Department Colloquium**

Wake Forest University, Winston-Salem, NC

April 6, 2016**Department Colloquium**

Winthrop University, Rock Hill, SC

April 1, 2016**re:boot Number Theory 2016**(co-organizer)

Duke University, Durham, NC

March 17-20, 2016**30th Automorphic Forms Workshop**

Wake Forest University, Winston-Salem, NC

March 7-10, 2016

### 2015

**Computational Aspects of the Langlands Program**(invited speaker)

ICERM, Providence, Rhode Island

November 8-21, 2015**Elementary, analytic, and algorithmic number theory: Research inspired by the mathematics of Carl Pomerance**(invited speaker)

University of Georgia, Athens, GA

June 9-11, 2015**Analytic Methods in Elementary Number Theory**(invited speaker)

AMS Special Session Southeastern Sectional

University of Alabama, Huntsville, AL

March 27-29, 2015**Department Colloquium**

Williams College, Williamstown, MA

January 23, 2015**UNC-Duke Number Theory Seminar**

Duke University, Durham, NC

January 21, 2015**2015 Joint Mathematics Meetings**

San Antonio, TX

January 10-13, 2015

### 2014

**New Approaches in Probabilistic and Multiplicative Number Theory**

CRM, Montréal, Canada

December 8-12, 2014**Bounded Gaps Between Primes**

AIM, Palo Alto, CA

November 17-21, 2014**Young Mathematicians Conference**

Ohio State University, Columbus, Ohio

August 22-24, 2014**ENFANT & ELEFANT**

Hausdorff Center, Bonn, Germany

July 11-18, 2014

**Letters of Recommendation – **Please read this note if you would like me to write you a letter of recommendation.

My research interests lie in the field of analytic number theory. In particular, I study properties of L-functions (such as the Riemann zeta-function, Dedekind zeta-functions, Artin L-functions, and automorphic L-functions), their zeros, and class groups of number fields.

My work is partially supported by the National Science Foundation under grant NSF CAREER DMS-2239681. It was previously supported by NSF DMS-1902193 (2019-2023, PI) and NSF FRG DMS-1854398 (2019-2023, Senior Scientist Consultant).

## Preprints

- Twisted 2k-th moments of primitive Dirichlet
*L*-functions: beyond the diagonal, with S. Baluyot, submitted, [preprint]

## Publications

- An unconditional Montgomery Theorem for pair correlation of zeros of the Riemann zeta function, with S. Baluyot, D.A. Goldston, and A. I. Suriajaya, to appear,
*Acta Arithmetica*[preprint] - On the Montgomery-Odlyzko method regarding gaps between zeros of the zeta-function, with D. A. Goldston and T. S. Trudgian, in press,
*Journal of Mathematical Analysis and Applications*, 2023 [preprint] [journal] - Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function, with T. Trudgian and A. Simonič,
*Transactions of the American Mathematical Society*,**375**, no. 3 (2022), 3239-3265. [preprint] [journal] - Hilbert transforms and the equidistribution of zeros of polynomials, with E. Carneiro, M. K. Das, A. Florea, A. V. Kumchev, A. Malik, M. B. Milinovich, and J. Wang,
*Journal of Functional Analysis*,**281**(2021), no. 9, Paper No. 109199, 25 pp. [preprint] [journal] - On a conjecture for l-torsion in class groups of number fields: from the perspective of moments, with L. B. Pierce and M. Matchett Wood,
*Mathematical Research Letters*,**28**, no. 2 (2021), pp. 575-621. [preprint] [journal] - An effective Chebotarev density theorem for families of number fields with an application to l-torsion in class groups, with L. B. Pierce and M. Matchett Wood,
*Inventiones Mathematicae*,**219**(2020), no. 2, 701-778. [preprint] [journal] - Extremal primes of elliptic curves without complex multiplication, with C. David, A. Gafni, A. Malik, and N. Prabhu,
*Proceedings of the American Mathematical Society*,**148**(2020), no. 3, 929-943. [preprint] [journal] - On
*r*-gaps between zeros of the Riemann zeta-function, with J. B. Conrey,*Bulletin of the London Mathematical Society*,**50**(2018) no. 2, 349-356. [preprint] [journal] - Gaps between zeros of Dedekind zeta-functions of quadratic number fields. II, with H.M. Bui and W.P. Heap,
*Quarterly Journal of Mathematics*,**67**(2016) no. 3, 467-482. [preprint] [journal] - Consecutive primes in tuples, with W. D. Banks and T. M. Freiberg,
*Acta Arithmetica*,**167**(2015), 261-266. [preprint] [journal] - Gaps between zeros of Dedekind zeta-functions of quadratic number fields,
*Journal of Mathematical Analysis and Applications*,**418**(2014), no. 1, 100-107. [journal] - Moments of products of automorphic L-functions, with M. B. Milinovich,
*Journal of Number Theory*,**139**(2014), 175-204. [journal]

## Collaborations with Undergraduates at the SMALL 2014 REU

### Publications

- Gaps between zeros of GL(2) L-functions, with O. Barrett, B. McDonald, S. J. Miller, P. Ryan, and K. Winsor,
*Journal of Mathematical Analysis and**Applications*,**429**(2015), 204-232. [preprint] [journal] - Gaussian distribution of the number of summands in generalized Zeckendorf decompositions in small intervals, with A. Best, P. Dynes, X. Edelsbrunner, B. McDonald, S. J. Miller, K. Tor, and M. Weinstein,
*INTEGERS,***16**(2016), #A6. [preprint] [journal]

### Conference Proceedings

- Benford behavior of generalized Zeckendorf decompositions, with A. Best*, P. Dynes, X. Edelsbrunner, B. McDonald, S. J. Miller, and M. Weinstein,
*Combinatorial and Additive Number Theory II: CANT*, New York, NY, USA, 2015 and 2016, Springer, New York, 2017. [preprint] [article] - Gaussian behavior of the number of summands in Zeckendorf decompositions in small intervals, with A. Best, P. Dynes, X. Edelsbrunner, B. McDonald, S. J. Miller, K. Tor, and M. Weinstein,
*Fibonacci Quarterly,***52**(2014), no. 5, 35-46. [preprint] [article] - Benford behavior of Zeckendorf decompositions, with A. Best, P. Dynes, X. Edelsbrunner, B. McDonald, S. J. Miller, K. Tor, and M. Weinstein,
*Fibonacci Quarterly,***52**(2014), no. 5, 47-53. [preprint] [article]

### Book Contributions

- From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond, with O. Barrett, F. W. K. Firk, and S. J. Miller, Chapter in Open Problems in Mathematics (editors John Nash Jr. and Michael Th. Rassias), Springer-Verlag (2016). [preprint] [chapter]
- Some Results in the Theory of Low-lying Zeros, with B. Mackall, S. J. Miller, C. Rapti, and K. Winsor, in Families of automorphic forms and the trace formula (editors Werner Müller, Sug Woo Shin, and Nicolas Templier), Simons Symposia Series, Springer-Verlag. [preprint] [chapter]
- Benfordness of the Cauchy Distribution and Generalizations, with X. Edelsbrunner, K. Huan, B. Mackall, J. Powell, and M. Weinstein, Section 3.6, an appendix to Benford’s Law: Theory and Applications, (S. J. Miller, editor), Princeton University Press, 2015. [chapter]

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