Zeros of the Riemann Zeta-function on the critical line.

D.R. Heath-Brown

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

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On large values of the Riemann zeta-function on short segments of the critical line

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We obtain a series of new conditional lower bounds for the modulus and the argument of the Riemann zeta function on very short segments of the critical line, based on the Riemann hypothesis. In particular, we prove that for any large fixed constant A > 1 there exist(non-effective) constants T₀(A) > 0 and c₀(A) > 0 such that the maximum of |ζ (0.5+it)| on the interval (T-h,T+h) is greater than A for any T > T₀ and h = (1/π)lnlnln{T}+c₀.