Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals

We examine I(sigma, T) = [GRAPHICS] dt for sigma = 1/2 + a/log T and a > 0 either constant or tending to zero w ith T. Assuming the Riemann Hypothesis, we obtain a formula for this mean v alue which depends on the pair correlation of zeros of the zeta-function. O n the Riemann Hypothesis alone this provides upper and lower bounds of the same order of magnitude for I(sigma, T), and assuming the pair correlation conjecture we obtain an asymptotic formula. Conversely, an asympotic formul a for I(sigma, T) implies the pair correlation conjecture. We also obtain t he equivalence of the asymptotic formula for I(sigma, T) with an asymptotic formula for an integral of Selberg connected with primes in short interval s.