Under the assumption of the Riemann Hypothesis, we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.
"Explicit relations between primes in short intervals and exponential sums over primes." Funct. Approx. Comment. Math. 51 (2) 379 - 391, December 2014. https://doi.org/10.7169/facm/2014.51.2.9