A generalization of the conjecture
$$\pi(x+x^{\theta}) - \pi(x) \sim \frac{x^\theta}{\log x} $$ (Ingham, 1937 or earlier) might be
$$\Delta \pi_k = \pi_k((x+1)^2) - \pi_k(x^2)\sim \frac{x}{\log x}\frac{(\log\log x^2)^{(k-1)}}{(k-1)!} $$
in which $\pi_k(x)$ is the number of numbers with k primes including repetitions not exceeding x.
For the case $k = 1$ we have $\Delta \pi_1 \sim \frac{x}{\log x},$ suggesting that the number of primes on a square interval approaches that of the interval $[1,x].$ All very speculative. My question is simply whether we can find a similar (speculative) statement for $\Delta \pi_2,$ etc., relating the number of primes on the square interval to that of some interval $[1,x]?$ In other words, can we solve for example:
$$\frac{x}{\log x}\log\log (x^2)= \frac{y}{\log y} \sim \pi(y) $$ for y in terms of x?
It's easy enough to plot $f(x)=\frac{x}{\log x}\log\log x$ and find $\pi(y)\approx f(x).$ For example:
$$\frac{1000}{\log 1000}\log\log 1000^2 \approx \frac{3048}{\log 3048}.$$
Thanks for any insight.
I have failed the Turing test twice today, so editing appreciated.
The solution $ > e$ of the equation $y/\log(y) = z$ is $y = - z\; \rm{LambertW}(-1,-1/z)$ (in Maple's terminology) for $z > e$.