Prime gaps and gaps between successive critical zeros of zeta

Question

Assuming RH, the sequence of critical zeros of the Riemann zeta function can be viewed as the Fourier transform of the sequence of primes. From a physicist point of view, the average gap between the elements of a sequence of points on a straight line is the reciprocal of the average gap of the sequence obtained through the application of the Fourier transform, up to a normalisation factor of $ 2\pi $. Is there thus a heuristics suggesting that the proportion of integers n below x such that $a\leq\dfrac{p_{n+1}-p_{n}}{\log p_{n}} \leq b$ is equal to the proportion of critical Riemann zeros $ 1/2+i\gamma_{n} $ of imaginary part less than T such that $\dfrac{2\pi}{b}\leq (\gamma_{n+1}-\gamma_{n})\log\gamma_{n}\leq\dfrac{2\pi}{a} $ with $ 0\lt a\lt b\lt\infty $?