We will show that the number of integers $\leq x$ 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 $\frac{ \log\log x}{ \log 2}$.
Categories: math.NT