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

Usually Riemann hypothesis is introduced along this lines.
(1.1) Geometric progressions were known since forever
(1.2) Euler factorization links a product of primes and a sum of natural numbers
(1.3) Harmonic series diverges, thus there are infinity many primes
(1.4) Riemann expanded zeta function to whole complex plane and conjured that all non-trivial zeroes have $\Re=1/2$

So far so good. Nice and steady. How did he come to this conclusion isn't obvious, but this is why RH is an open problem. Suddenly...

(2.1) Non-trivial zeroes of zeta function impose constraints on distribution of prime numbers including the accuracy of prime counting function distribution of (a) gaps between primes, (b) twin primes ...

Indeed, (1.2) linked all natural numbers to primes. But why distribution of zeroes is relevant? Why not Fourier transform of zeta function, not distribution of ones?

In other words, I understand why a certain property of zeta function is relevant to primes. But why this certain property happen to be the distribution of zeroes?

Edit: I've read this What is so interesting about the zeroes of the Riemann $\zeta$ function? but roughly around this point

"This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their 'expected' positions."

"Roughly speaking, the explicit formula says the Fourier transform of the zeros of the zeta function is the set of prime powers plus some elementary factors."

it gets hard to follow.

• @Peter Sheldrick, Rigorous proofs are tough, but I am asking about the line of a sketch at a "5K basic complex analysis" level. – sixtytrees Jul 12 '16 at 16:28
• the Abel summation formula gives $\ln \zeta(s) = \sum_{p^k} \frac{p^{-sk}}{k} = s \int_2^\infty J(x) x^{-s-1}dx$ where $J(x) = \sum_{p^k < x} \frac{1}{k} = \pi(x)+ \mathcal{O}(\sqrt{x})$. With the change of variable $x = e^u$ you get $\frac{\ln \zeta(s)}{s} = \int_{\ln 2}^\infty J(e^u) e^{-su} du$ i.e. the Laplace/Fourier transform of $J(e^u)$ – reuns Sep 11 '16 at 8:34
• Good post. Just a comment from a non professional, is that $$\log n=\sum_{d\mid n}\Lambda(d).$$ I've thought a lot this week about what impact could have the non existence of infinitely many twin primes. I agree that the question is strange: because there is only a definition of logarithm (one can see this definition in a book of analysis), but there is some that I don't understand. – user243301 Sep 11 '16 at 12:54
• And Riemann found that the functional equation meant $\xi(s) = s (s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s)$ obeys to $\xi(s) = \xi(1-s)$. Together with $\xi(s) = \overline{\xi(\overline{s})}$ it means $\Xi(t) = \xi(1/2+it)$ is real. Proving it changes of sign infinitely many times is not difficult, so that it has infinitely many zeros $\implies \zeta(s)$ has infinitely many zeros on $Re(s) = 1/2$. This is the main reason why he made his hypothesis. – reuns Sep 11 '16 at 13:07

• with $\psi(x) = \sum_{p^k < x} \ln p$ you have the Riemann explicit formula $$\psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} + \mathcal{O}(1)$$ ($\rho$ being the non trivial zeros of $\zeta(s)$, obtained from the residue theorem applied to $\frac{-\zeta'(s)}{\zeta(s)} = \sum_{p^k} p^{-sk}\ln p$ $= s \int_2^\infty \psi(x) x^{-s-1}dx$ $\implies \psi(x) = \frac{1}{2i\pi} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{-\zeta'(s)}{\zeta(s)} \frac{x^s}{s}ds$ by inverse Laplace/Mellin transform)
• then the Riemann hypothesis is that $$\frac{\psi(e^u) - e^u}{e^{u/2}} = - \sum_\rho \frac{e^{i u \text{ Im}(\rho)}}{\rho} + \mathcal{O}(e^{-u/2})$$ i.e. $\frac{\psi(e^u) - e^u}{e^{u/2}}$ is almost periodic, having an expansion that is "almost" a Fourier series (you doubt it would have a huge impact on the distribution of prime numbers)