Student Analytic Number Theory

An old result of Nathanson shows that every n-element set of positive reals has at least n(n+1)/2+1 distinct subset sums, with equality exactly for homogeneous arithmetic progressions. We establish stability versions of this inverse theorem in two regimes. First, for any parameter M at most n-4…

Zaremba's famous conjecture (1972) arose from the theory of numerical integration and relates to the field of continued fractions. It predicts that for any given prime p there is a positive integer a < p such that when expanded as a continued fraction a/p = 1/c_1+1/c_2 +... + 1/c_s all…

The Steinhaus function is a random, completely multiplicative function on the integers, whose values on primes are i.i.d random variables uniformly distributed on the complex unit circle. Its study is motivated by the study of deterministic multiplicative functions such as the Möbius function…

The Lonely Runner Conjecture, due to Wills and Cusick, asserts that if n runners with distinct constant speeds run around a unit length track, all starting at a common point, then each runner is at some moment separated by a distance of at least 1/n from every other runner. 

A weaker…

The Bateman-Horn conjecture predicts the density of primevalues assumed by integer polynomials. In this talk, we discuss recentprogress on an average version of this conjecture, achieved by employingHooley neutralizers in combination with standard sieving techniques. Inparticular, our findings…

The prime number 357686312646216567629137 is notable in some recreational math circles because of the unusual property that it remains prime successively on removing the left digit until there are no remaining digits. In this talk we will explore the more general phenomena of prime truncations…

A typical Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, and are so “classical” that sometimes practitioners…

Among the nondegenerate C^4 hypersurfaces, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. For all other hypersurfaces, we give a heuristically sharp lower bound for the number of rational points near them, improving the…

In 2023, Yueke Hu and Paul Nelson, in their paper ``Subconvex bounds for $U_{n + 1} \times U_n$ in horizontal aspects'', established a subconvex bound valid in certain horizontal aspects for $L$-functions attached to automorphic representations of unitary groups $U_{n + 1} \times U_n$. My…

The polynomial Szemer\'edi theorem of Bergelson and Leibman gives broad conditions under which polynomial patterns must appear in every positive-density subset of Z^d. When the polynomials do not vanish at zero, the correct replacement condition is conjecturally joint intersectivity, a…