Fig 3. Prime numbers.

Left panel: In red the map x′ = x/(1 − x ln x) (f*(x) = x exp_z(u|x|^z−1 ln x) with z = 2 and u = 1). In blue the trajectory $x_t=h^{-1}(-t),h\left(x\right)={\int }_{-1/2}^{x}dx(1/x^2\text{ln}\left|x\right|)$ initiated at x₀ = −1/2. Its positions approximate asymptotically (minus) the reciprocals of Prime numbers. Right panel: In red the map x′ = x + ln x (f*(x) = x exp_Z(u|x|^Z−1 ln x) with Z = 0 and u = 1). In blue the trajectory $x_t={\text{Li}}^{-1}\left(t\right),\text{Li}\left(x\right)={\int }_2^{x}dx1/\text{ln}\left|x\right|$ initiated at x₀ = 2. Its positions approximate asymptotically the Prime numbers. Identity lines in green.