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ON THE VALUE DISTRIBUTION OF THE ZETA-FUNCTION ON THE CRITICAL LINE M. JUTILA 1. Introduction Determining the order of magnitude of Riemann's zeta-function £(s) on the critical line Res = 7 is a very difficult problem, even on the assumption of the Riemann hypothesis. The object of this note is to show that the corresponding "statistical" problem can be essentially solved. We are interested in estimates of l(("2 + '0l which hold with probability one; in other words, the measure of the subset of the t-interval [0, T] , where the estimate fails to be valid, should be o{T) as T tends to infinity. Such an estimate is given in the corollary of Theorem 1, and Theorem 2 shows that the result is in a logarithmic sense best possible. The constants implied by the symbols <^ , ^>, and O(...), as well as the other constants to be introduced, will be always absolute unless otherwise indicated. THEOREM 1. Let T ^ 2, 1 ^ V ^ log T, and denote by M {V) the set of the numbers t e [0, T] such that > V. (1.1) Then the measure of the set M (V) satisfies loglog T and
Bulletin of the London Mathematical Society – Wiley
Published: Sep 1, 1983
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