On simple zeros of the Riemann zeta-function in short intervals on the critical line

We calculate in a new way (following old ideas of Atkinson and new ideas of Jutila and Motohashi) the mean square of the product of a function F(s), involving the Riemann zeta-function zeta(s), and a certain Dirichlet polynomial A(s) of length M = T-theta in short intervals on sigma = a near the critical line: if 0 < 3/8, then integral(T)(T+H) \AF(a +it)\(2) dt = I(T, H) + O(T1/3+epsilon M-4/3). The main term I(T, H) is well known, but the error term is much smaller than the one obtained by other approaches (e.g. O(T1/2+epsilon M)). It follows from Levinson's method that the proportion of zeros of the zeta-function with imaginary parts in [T, T + H] which are simple and on the critical line is positive, when H greater than or equal to T-0.552.