Explicit relations between primes in short intervals and exponential   sums over primes
  Under the assumption of the Riemann Hypothesis (RH), 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. 
Alessandro Languasco and Alessandro Zaccagnini
math.NT