A consequence of the RH is
$$|\pi(x)-\mathrm{li}(x)|<\frac{1}{8\pi}\sqrt{x}\log x\quad\forall x\geq 2657$$
where $\mathrm{li}(x)$ is the logarithmic integral, $\pi(x)$ is the prime counting function and $\log x$ is the natural logarithm. Another consequence is
$$|\psi(x)-x|<\frac{1}{8\pi}\sqrt{x}\log^2 x\quad\forall x\geq73.2$$
Where $\psi(x)$ is Chebyshev's second function (not to be confused with the digamma function). Yet another implication is that forall $x\geq 2$ there is a prime $p$ satisfying
$$x-\frac{4}{\pi}\sqrt{x}{\log x}<p\leq x$$
If the riemann hypothesis is true, then the gap between a prime $p$ and its successor prime is $O(\sqrt{p}\log p)$.
The riemann hypothesis implies
$$-\sum_{k=1}^{\infty}\frac{(-x)^k}{(k-1)!\zeta(2k)}=O\left(x^{\frac{1}{4}+\epsilon}\right)\text{ holds }\forall\epsilon>0$$
Source:Wikipedia
The grand riemann hypothesis implies
$$\lim_{x\to 1^{-}}\sum_{p\geq 2}(-1)^{(p+1)/2}x^p=+\infty$$
Here is a good article about its consequences.
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