The multiplication table constant and sums of two squares

Abstract

We will show that the number of integers $\le x$ that can be written as the square of an integer plus the square of a prime equals $\frac{\pi }{2}\cdot \frac{x}{\log x}$ minus a secondary term of size $x/(\log x{)}^{1+\delta +o(1)}$, where $\delta :=1-\frac{1+\log \log 2}{\log 2}=0.0860713320\dots$ is the multiplication table constant. Detailed heuristics suggest that this secondary term is asymptotic to $\frac{1}{\sqrt{\log \log x}}\cdot \frac{x}{(\log x{)}^{1+\delta }}$ times a bounded, positive, $1$-periodic, non-constant function of $\frac{\log \log x}{\log 2}$.