Research
My primary research interests are in the applications of analytic number theory to arithmetic statistics, and often problems with concrete applications to arithmetic.
Selected publications
 An approximate form of the Artin holomorphy conjecture and nonvanishing of Artin Lfunctions. (With J. Thorner, A. Zaman)
Inventiones Mathematicae, 2023.
(journal, arxiv, code)
 Upper bounds on number fields of given degree and bounded discriminant. (With F. Thorne)
Duke Mathematical Journal, 2022.
(journal, arxiv)
 3isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field. (With M. Bhargava, Z. Klagsbrun, A. Shnidman)
Duke Mathematical Journal, 2019.
(journal, arxiv)
 Unexpected biases in the distribution of consecutive primes. (With K. Soundararajan)
Proceedings of the National Academy of Sciences, 2016.
(journal, arxiv, Quanta article)
Preprints
 The average size of 3torsion in class groups of 2extensions. (With J. Wang, M. Wood)
(arxiv)
 Normal distribution of bad reduction. (With D. Loughran, A. Shnidman)
(arxiv)
 Uniform exponent bounds on primitive extensions of number fields.
(arxiv, code)
 Faithful Artin induction and the Chebotarev density theorem. (With A. Smith)
(arxiv)
All published works (chronological)
Please note that published versions of these papers may differ slightly from the versions below.
 On the counting function for the generalized Niven numbers. (With R. Daileda, J. Jou, E. Rossolimo, and E. Treviño)
J. Théor. Nombres Bordeaux, 21 no. 3 (2009), pp. 503515.
(pdf)
 Proof of the AlderAndrews Conjecture. (With C. Alfes and M. Jameson)
Proc. Amer. Math. Soc., 139 no. 1 (2011), pp. 6378.
(pdf)
 On a conjecture of Andrews. (With M. Jameson)
Math. Res. Lett., 17 no. 6 (2010), pp. 11511154.
(pdf)
 Almostprimes represented by quadratic polynomials.
Acta Arith., 151 (2012), pp. 241261.
(pdf)
 Gauss sums over finite fields and roots of unity.
Proc. Amer. Math. Soc., 139 no. 4 (2011), pp. 12731276.
(pdf)
 Etaquotients and theta functions.
Adv. Math., 241 (2013), pp. 117.
(pdf)
 Pretentiously detecting power cancellation. (With J. Jung)
Math. Proc. Camb. Phil. Soc., 154 no. 3 (2013), pp. 481498.
(pdf)
 Multiplicative functions dictated by Artin symbols.
Acta Arith., 161 (2013), pp. 2131.
(pdf)
 Representation by ternary quadratic forms.
Bull. Lond. Math. Soc., 46 no. 6 (2014), pp. 12371247.
(pdf)
 The distribution of 2Selmer ranks of quadratic twists of elliptic curves with partial twotorsion. (With Z. Klagsbrun)
Mathematika.
(pdf)
 Bounded gaps between primes in number fields and function fields. (With A. Castillo, C. Hall, P. Pollack, and L. Thompson)
Proc. Amer. Math. Soc., 143 no. 7 (2015), pp. 28412856.
(pdf)
 The distribution of the Tamagawa ratio in the family of elliptic curves with a twotorsion point. (With Z. Klagsbrun)
Res. Math. Sci., 1:15.
(pdf, open access)
 The number of ramified primes in number fields of small degree. (With F. Thorne)
Proc. Amer. Math. Soc., 145 no. 8 (2017).
(pdf)
 Best possible densities of Dickson mtuples, as a consequence of ZhangMaynardTao. (With A. Granville, D. M. Kane, and D. Koukoulopoulos)
Analytic Number Theory, in honor of Helmut Maier's 60th Birthday, Springer (eds. C. Pomerance and M. Rassias).
(pdf)
 Effective logfree zero density estimates for automorphic Lfunctions and the SatoTate conjecture. (With J. Thorner)
IMRN
(pdf)
 Unexpected biases in the distribution of consecutive primes. (With K. Soundararajan)
PNAS, 113 no. 31 (2016).
(pdf, slides, code)
 The generalized NagellLjunggren problem: powers with repetitive representations. (With A. Bridy, A. Shallit, J. Shallit)
Exp. Math.
(pdf)
 The distribution of consecutive prime biases and sums of sawtooth random variables. (With K. Soundararajan)
Math. Proc. Camb. Phil. Soc.
(pdf)
 Threeisogeny Selmer groups and ranks of abelian varieties in quadratic twist families. (With M. Bhargava, Z. Klagsbrun, A. Shnidman)
Duke Math. J..
(pdf, code)
 Zeros of twisted Lfunctions near Re(s)=1. (With J. Thorner)
Appendix to Shimura curves and the abc conjecture by H. Pasten.
Journal of Number Theory, Prime
(arxiv, pdf)
 Rank growth of elliptic curves in nonabelian extensions. (With F. Thorne)
IMRN.
(pdf, slides)
 Elements of given order in TateShafarevich groups of abelian varieties in quadratic twist families. (With M. Bhargava, Z. Klagsbrun, A. Shnidman)
Algebra and Number Theory
(pdf,arxiv)
 Upper bounds on polynomials with small Galois group. (With F. Thorne)
Mathematika
(pdf,arxiv)
 Improved lower bounds for the number of fields with alternating Galois group. (With A. Landesman and F. Thorne)
BLMS
(pdf, arxiv)
 Upper bounds on number fields of given degree and bounded discriminant. (With F. Thorne)
Duke Math. J.
(pdf, arxiv, slides)
 Asymptotic identities for additive convolutions of sums of divisors. (With S. Shrestha, F. Thorne)
Math. Proc. Camb. Phil. Soc.
(pdf, arxiv)
 An approximate form of Artin's holomorphy conjecture and nonvanishing of Artin Lfunctions. (With J. Thorner, A. Zaman)
Inventiones Mathematicae
(pdf, arxiv)
 Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants. (With T. Anderson, A. Gafni, D. LowryDuda, G. Shakan, and R. Zhang)
IMRN
(pdf,arxiv)
 Improved bounds on number fields of small degree. (With T. Anderson, A. Gafni, K. Hughes, D. LowryDuda, F. Thorne, J. Wang, R. Zhang)
Discrete Analysis
(arxiv)
