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Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type 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 Fourier-like transform? Can this be used to get an analogue of Erdos-Kac'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 so-called "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.

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