Loosely speaking, Riemann's explicit formula states that there exists a Fouriertype duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the distribution of prime gaps is invariant under a Fourierlike transform? Can this be used to get an analogue of ErdosKac's theorem where $\omega(n)$ is replaced by $p_{n+1}p_{n}$ and its normal order $\log\log n$ by $\log n$ following the prime number theorem? If so, can one expect a proof (under Goldbach's conjecture) of the socalled "symmetric density conjecture" and "increasing density conjecture" considered in Would the following conjectures imply Cramer's conjecture? ? Could this also give further evidence for Montgomery's pair correlation conjecture?
Thanks in advance.
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


