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SIMPLE ZEROS OF THE RIEMANN ZETA-FUNCTION ON THE CRITICAL LINE D. R. HEATH-BROWN The Riemann hypothesis states that all non-real zeros of the Riemann Zeta-func- tion £(s) lie on the " critical line " Re (s) = %. This has never been proved or dis- proved, but it is known (see Brent [1]) that the first 40,000,000 zeros do in fact lie on the line Re (s) = •£. Moreover Levinson [2], [4] has shown that N (T)> 0-3474N(T) (1) for sufficiently large T, where, as usual, N(T) denotes the number of zeros of £(s) in the rectangle 0^(7<l,0<f < T, and N (T) denotes the number of zeros on the line segment a —\, 0 ^ t ^ T; in each case the zeros are counted according to multiplicity. Important though a proof of the Riemann hypothesis would be, it would still leave many unanswered questions concerning the " vertical" distribution of the zeros. The most fundamental question of this type is whether the zeros are all simple. This is indeed the case for the first 40,000,000 zeros, as shown by Brent [1]. Moreover Montgomery [5] has found, assuming the truth of the Riemann hypothesis, that N (T)
Bulletin of the London Mathematical Society – Wiley
Published: Mar 1, 1979
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