Robert J. Lemke Oliver

Research

My primary research interests are in the applications of analytic number theory to arithmetic statistics, and often problems with concrete applications to arithmetic. My publications are listed below, from oldest to newest. Please note that published versions of these papers may differ slightly from the versions below.

On the counting function for the generalized Niven numbers, J. Théor. Nombres Bordeaux, 21 no. 3 (2009), pp. 503-515. (With R. Daileda, J. Jou, E. Rossolimo, and E. Treviño) (pdf)
Proof of the Alder-Andrews Conjecture, Proc. Amer. Math. Soc., 139 no. 1 (2011), pp. 63-78. (With C. Alfes and M. Jameson) (pdf)
On a conjecture of Andrews, Math. Res. Lett., 17 no. 6 (2010), pp. 1151-1154. (With M. Jameson) (pdf)
Almost-primes represented by quadratic polynomials, Acta Arith., 151 (2012), pp. 241-261. (pdf)
Gauss sums over finite fields and roots of unity, Proc. Amer. Math. Soc., 139 no. 4 (2011), pp. 1273-1276. (pdf)
Eta-quotients and theta functions, Adv. Math., 241 (2013), pp. 1-17. (pdf)
Pretentiously detecting power cancellation, Math. Proc. Camb. Phil. Soc., 154 no. 3 (2013), pp. 481-498. (With J. Jung) (pdf)
Multiplicative functions dictated by Artin symbols, Acta Arith., 161 (2013), pp. 21-31. (pdf)
Representation by ternary quadratic forms, Bull. Lond. Math. Soc., 46 no. 6 (2014), pp. 1237-1247. (pdf)
The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion, Mathematika. (With Z. Klagsbrun) (pdf)
Bounded gaps between primes in number fields and function fields, Proc. Amer. Math. Soc., 143 no. 7 (2015), pp. 2841-2856. (With A. Castillo, C. Hall, P. Pollack, and L. Thompson) (pdf)
The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point, Res. Math. Sci., 1:15. (With Z. Klagsbrun) (pdf, open access)
The number of ramified primes in number fields of small degree, Proc. Amer. Math. Soc., 145 no. 8 (2017). (With F. Thorne) (pdf)
Best possible densities of Dickson m-tuples, as a consequence of Zhang-Maynard-Tao, Analytic Number Theory, in honor of Helmut Maier's 60th Birthday, Springer (eds. C. Pomerance and M. Rassias). (With A. Granville, D. M. Kane, and D. Koukoulopoulos) (pdf)
Effective log-free zero density estimates for automorphic L-functions and the Sato-Tate conjecture, IMRN. (With J. Thorner) (pdf)
Unexpected biases in the distribution of consecutive primes, PNAS, 113 no. 31 (2016). (With K. Soundararajan) (pdf, slides, code)
The generalized Nagell-Ljunggren problem: powers with repetitive representations, Exp. Math. (With A. Bridy, A. Shallit, J. Shallit) (pdf)
The distribution of consecutive prime biases and sums of sawtooth random variables, Math. Proc. Camb. Phil. Soc. (With K. Soundararajan) (pdf)
Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families, Duke Math. J.. (With M. Bhargava, Z. Klagsbrun, A. Shnidman) (pdf, code)
Zeros of twisted L-functions near Re(s)=1. (With J. Thorner) Appendix to Shimura curves and the abc conjecture by H. Pasten. (arxiv, pdf)
Rank growth of elliptic curves in nonabelian extensions, IMRN. (With F. Thorne) (pdf, slides)
Elements of given order in Tate--Shafarevich groups of abelian varieties in quadratic twist families, Algebra and Number Theory. (With M. Bhargava, Z. Klagsbrun, A. Shnidman) (pdf,arxiv)
Upper bounds on polynomials with small Galois group, Mathematika. (With F. Thorne) (pdf,arxiv)
Improved lower bounds for the number of fields with alternating Galois group, BLMS. (With A. Landesman and F. Thorne) (pdf, arxiv)
Upper bounds on number fields of given degree and bounded discriminant, Duke Math. J. (With F. Thorne) (pdf, arxiv, slides)
Asymptotic identities for additive convolutions of sums of divisors, Math. Proc. Camb. Phil. Soc., (With S. Shrestha, F. Thorne) (pdf, arxiv)
An approximate form of Artin's holomorphy conjecture and non-vanishing of Artin L-functions, submitted. (With J. Thorner, A. Zaman) (pdf, arxiv)
Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants, IMRN. (With T. Anderson, A. Gafni, D. Lowry-Duda, G. Shakan, and R. Zhang) (pdf,arxiv)
The average size of 3-torsion in class groups of 2-extensions, submitted. (With J. Wang, M. Wood) (pdf,arxiv,slides)
Improved bounds on number fields of small degree, submitted. (With T. Anderson, A. Gafni, K. Hughes, D. Lowry-Duda, F. Thorne, J. Wang, R. Zhang) (arxiv)

My Ph.D. thesis was creatively titled Topics in analytic number theory and consisted of papers 2-9 above, together with a unifying introduction. If you'd like to read it, you may do so here.