Consequences of the Riemann hypothesis I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice to include consequences of the generalized Riemann hypothesis (but specify which one is assumed).
 A: As for GRH, the prettiest one I know is this complete solution of the odd Goldbach conjecture (that every  number greater than 5 is a sum of 3 primes).
Deshouillers, J.-M.; Effinger, G.; te Riele, H.; Zinoviev, D., A complete Vinogradov 3-primes theorem under the Riemann hypothesis, Electron. Res. Announc. Am. Math. Soc. 3, 99-104 (1997). ZBL0892.11032.
A: (Jeffrey C. Lagarias) The following is equivalent to RH. Let $H_n = \sum\limits_{j=1}^n \frac{1}{j}$
be the $n$-th harmonic number. For each $n \ge 1$
$$\sum\limits_{d\mid n} d \le H_n + \exp (H_n) \log (H_n),$$
with equality only for $n = 1.$
(An Elementary Problem Equivalent to the Riemann Hypothesis. See also OEIS A057641.)
A: From Wikipedia's page on the consequences of the Riemann hypothesis
"Riemann's explicit formula for the number of primes less than a given number in terms of a sum over the zeros of the Riemann zeta function says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. [...] the Riemann hypothesis is equivalent to the "best possible" bound for the error of the prime number theorem."
A: I haven't seen this application mentioned:
The closed horocycle $\gamma_l$ of length $l$ of the modular surface equidistributes to the hyperbolic volume form $\omega$ when $l\to +\infty$, and RH is equivalent to the error term:
$$\frac{1}{l} \int_{\gamma_l} f = \int f \omega  + o(l^{-3/4+\epsilon})$$
for every smooth function with compact support.
This is due to Don Zagier. https://people.mpim-bonn.mpg.de/zagier/files/scanned/EisensteinRiemannZeta/eisenstein-zeta-978-3-662-00734-1_10.pdf
Verjovsky has another reformulation of the Riemann hypothesis in terms of convergence of measures. https://projecteuclid.org/journals/kodai-mathematical-journal/volume-17/issue-3/Discrete-measures-and-the-Riemann-hypothesis/10.2996/kmj/1138040054.full
A: Many class group computations are sped up tremendously by assuming the GRH. As I understand it this is done by computing upper bounds on the discriminants of potential abelian extensions. See this survey by Odlyzko for more details
http://archive.numdam.org/ARCHIVE/JTNB/JTNB_1990__2_1/JTNB_1990__2_1_119_0/JTNB_1990__2_1_119_0.pdf
This is built into SAGE.
sage: J=JonesDatabase()
sage: NFs=J.unramified_outside([2,3])
sage: time RHCNs = [K.class_number(proof=False) for K in NFs]
CPU times: user 7.05 s, sys: 0.07 s, total: 7.13 s
Wall time: 7.15 s
sage: time CNs = [K.class_number() for K in NFs]
CPU times: user 20.19 s, sys: 0.24 s, total: 20.43 s
Wall time: 20.96 s

A: 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.
