Riemann hypothesis and prime distribution

Question

What exactly does the Riemann hypothesis imply for the prime numbers? Since the explicit formula is independent of the Riemann hypothesis, what would it actually mean for the primes if all the nontrivial zeros of the zeta function had real part 0.5? Is there some sort of a "simple" explanation for that?

Answer 1

The Riemann hypothesis says that for any real number $x$ the number of prime numbers less than $x$ is approximately $\mathrm{Li}(x)$ and this approximation is essentially square root accurate. More precisely, $$ \pi(x)=\mathrm{Li}(x)+O(\sqrt{x}\log(x)). $$

"Von Koch (1901) proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem."

References at this site:

How related is the distribution of primes to the Riemann Hypothesis?

What is the link between Primes and zeroes of Riemann zeta function?

Answer 2

Thanks Dietrich Burde, I've checked the links you mentioned and in the first response to the first link (How related is the distribution of primes to the Riemann Hypothesis?) it says: "Knowledge of the real part of the location of the zeta zeros translates into knowledge of the distribution of primes." Is it possible to clarify that "knowledge"? What do I know if I assume RH? And how exactly do I gain this knowledge?

Answer 3

The key link here is the explicit formula $$ \psi(x)=x-\sum_{\rho} \frac{x^{\rho}}{\rho}-\frac{\zeta'(0)}{\zeta(0)}-\frac{1}{2}\log\Big(1-\frac{1}{x^2}\Big)\ , $$ where $\sum_{\rho}$ denotes summing over all the zeroes $\rho$ of $\zeta$ with $0<\text{Re}(\rho)<1$ and $$\psi(x)=\sum_{p^k\leq x,\ p\text{ prime}}\log p$$ is the second Chebyshev function. This is proved using complex analysis, contour integration and Rouche's theorem, which links the zeroes of a function $f$ to an integral involving $f'/f$.

Now, it can be shown that if RH is true, then the sum $\sum_{\rho}$ in the explicit formula can be controlled to keep it small (because $\text{Re}(\rho)=1/2$ for all $\rho$) so that we get $$\psi(x)=x+O\big(\sqrt{x}(\log x)^2\big),$$ which in turn can be used to show that $$\pi(x)=\text{Li}(x)+O\big(\sqrt{x}\log x\big).$$

You can try reading Ram Murty's Problems in Analytic Number Theory for more details.