I'm a beginner. When I searched for results in probabilistic number theory most of the results were asymptotic in nature. Are there any results like with probability 1-$\epsilon$ (w.h.p) some property $P(\epsilon)$ hold? Any references would be also helpful.



Non-asymptotic results in probabilistic number theory
[Chebyshev's bias](https://en.wikipedia.org/wiki/Chebyshev's_bias) says that there are slightly more non-Pythagorean primes than Pythagorean primes (although the limiting frequency is the same).


