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In this paper we show that at least 2/5 of the zeros of the Riemann zeta-function are simple and on the critical line. Our method is a refinement of the method Levinson [11] used when he showed that at least 1/3 of the zeros are on the critical line (and are simple, äs observed by Heath-Brown [10] and, independently, by Seiberg). The main new element here is the use of a mollifier of length y=T with = 4/7 — whereas in Levinson's theorem the mollifier has 0 = 1/2 — . The work [6] of Deshouillers and Iwaniec on averages of Kloosterman sums is what allows us to use a longer mollifier. In fact, in their paper [7], they obtain an upper bound of essentially the right magnitude for an integral of the modulus squared of the zeta function multiplied by a mollifier of length T.

More than two fifths of the zeros of the Riemann zeta function are on the critical line.