GRH = Generalized Reiman Hypothesis.

I’ve always kind of had the vibes that “assuming GRH, then any reasonable randomness statement you could want to make about the primes is true”.

It turns out GRH is a bit more modest then that, which maybe highlights just how little we know about numbers.

But anyways, I looked around at SO and wikipedia and will summarize what I feel seem like some important consequences of GRH.

1. GRH implies that there is a generator for \(\mathbb{Z}_p^{\times}\) of size at most \((\ln(p))^{6}\).

   Also that \(\left\{ x< 2(\ln n)^{2}\right\}\) generates \(\mathbb{Z}_n^{\times}\).

2. GRH implies that prime gap is like \(\sqrt{p}\log p\)

note that I think we already know this if you replace \(\sqrt{p}\) with \(p^{\frac{5}{8}}\) so this one feels less important

3. The number of primes in the arithmetic progression $a, a+d, a+2d, , $ (where \(a\perp d\)) is like \(\frac{1}{\phi(d)} \frac{x}{\ln x} \pm \sqrt{x}\)

Furthermore, the first prime \(\equiv a \mod n\) is of size at most \(O((n\log n)^{2})\)

note that we already know this for if you replace \(n^{2}\) with likke \(n^{6}\) so again, doesn’t seem that pressing.

4. if you take a character sum over some interval, e.g., count the number of quadratic residues in some interval of length \(N\) then we already know its like bounded by \(\sqrt{p}\log p\) but under GRH you can replace the log with loglog. Again, not super impressive.

So in summary, I think (1) is the only one where assuming GRH really gets you something rather extremely good. huh, that’s really too bad. I wanted to prove hashing conditional on GRH. but this now seems very unlikely to be super helpful.