On simple zeros of the Riemann zeta-function

We investigate the distribution of simple zeros of the Riemann zeta-function. Let H ≤ T and L = log T . 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 F (s) = ζ(s) + 1 Lζ ′(s) and a certain Dirichlet polynomial A(s) = ∑ n≤M a(n) ns of length M = T θ with θ < 38 near the critical line: if R is a positive constant, a = 12− R L and a(n) = μ(n)n a− 1 2 ( 1− logn logM ) , then ∫ T+H T |AF (a+ it)| dt = H ( 1 2 + θ 6 ( 1−R− 1 2R ) − 1 2Rθ ( 1 + 1 R + 1 2R2 ) +e ( θ 12R + 1 4R3θ ) + o(1) ) +O ( T 1 3 M 4 3 ) . The main term is well known, but the error term is much smaller than the one obtained by other approaches (e.g. O ( T 1 2 M ) ). It follows from Levinson’s method, with an appropriate choice of R, that a positive proportion of the zeros of the zeta-function with imaginary parts in [T, T +H] lie on the critical line and are simple, when H ≥ T 0.591 (and by an optimal but more complicated choice of A(s) even when H ≥ T 0.552)! For shorter intervals we find with the Method of Conrey, Ghosh and Gonek ∑ T<γ≤T+H ζ ′(%) = HL2 4π +O ( HL+ T 1 2 +ε ) , where the sum is taken over the nontrivial zeros % = β + iγ of ζ(s). So every interval [T, T + T 1 2 ] contains the imaginary part of a simple zero of ζ(s)! Hence ] { % : T < γ ≤ T +H, ζ ′(%) 6= 0 } HT− 12−ε. With a density result of Balasubramanian we get even a nontrivial restriction for the real parts: e.g. at the limit of our results with Levinson’s method we find simple zeros % = β+iγ of the zeta-function with T < γ ≤ T + T 0.55 and 12 ≤ β ≤ 41 42 + ε.