The conjectural least-common-multiple asymptotic for x² + y² + 1

Let $F=x^2+y^2+1$, and define

$$\psi_F(N)=\log\left(\mathop{\operatorname{LCM}}_{n\leq N}\left\lbrace F(n)\right\rbrace\right).$$

Let $C_k$ be the conjectural constants for representations $kp=x^2+y^2+1$, and let

$$L=\frac{1}{\sqrt{2}}\prod_{p\equiv3\bmod4}\left(1-\frac{1}{p^2}\right)^{-1/2}.$$

The conjecture for $\psi_F$. One should have

$$\psi_F(N)\sim c_F\frac{N\log\log N}{\sqrt{\log N}},$$

where

$$c_F=\lim_{N\to\infty}\frac{1}{\log N}\sum_{k\leq N}C_k=L\approx0.76422\dots.$$

The paper proves the order of magnitude $N\log\log N/\sqrt{\log N}$ under its hypotheses, but precise asymptotics and the displayed constant are beyond the methods used; this prediction depends on the preceding conjectures.