Abstracts for Conference on "Asymptotic Counting and L-Functions"

Let $r_1(n)$ be the number of representations of n as the sum of a square and a square of a prime. We discuss the erratic behavior of $r_1$, which is similar to the one of the divisor function. We will show that the number of integers up to x that have at least one such representation is asymptotic to $(\pi/2) x / \log x$ minus a secondary term of size $x/(\log x)^{1+d+o(1)}$, where $d$ is the multiplication table constant. Detailed heuristics suggest very precise asymptotic for the secondary term as well.

It has been known for more than a century that many classical problems in analytic number theory would follow from a deeper understanding of partial sums and correlations of multiplicative functions. In recent years, this subject has seen several transformative breakthroughs, leading to numerous applications in number theory, ergodic theory, and combinatorics. The aim of the talk is to discuss some of the ideas behind these developments.

In this talk we discuss prime components and thickened prime components in Apollonian circle packings. In particular, we are interested in the set of curvatures that appear in these subsets of Apollonian circle packings and we prove first lower bounds on the number of curvatures of bounded height that appear in a thickened prime component. This is joint work with Elena Fuchs, Holley Friedlander, Piper Harris, Catherine Hsu, James Rickards, Katherine Sanden and Katherine Stange. 

The Euler-Kronecker constant of a number field $K$ is defined as the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function associated with $K$ at $s=1$. Investigating the distribution of the normalized difference of the Euler-Kronecker constants of the prime cyclotomic field $\mathbb{Q}(\zeta_q)$ and its maximal real subfield, we connect it with Kummer's conjecture. This conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields.