The multiplication table constant and sums of two squares 

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. In particular, our proofs imply that the main contribution to the mean value of $r_1(n)$ comes from integers with “unusual” number of prime factors, i.e, those with $\omega(n) \sim 2 \log \log x$ (for which $r_1(n) \sim (\log x)^{\log 4-1})$, where $\omega(n)$ is the number of district prime factors of n.   

In the talk we will review the results of several works that include a recent joint preprint with Andrew Granville and Cihan Sabuncu and my paper from 2022 as well as some work in progress.