Zeros of the Derivatives of the Riemann Zeta-function ∗

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with the mollifier B(s) introduced in Theorem A. For the error term O(T/L), we need some condition of gj that will be discussed at the end of the proof. We note that simple zeros of ξ(s) are not zeros of f(s), besides multiple zeros of ξ(s) are zeros whose multiplicities decrease by one. From symmetry of zeros of ξ(s) to 1/2, the left hand side of (4.4) is not less than R L (N(T ) − Nd(T )). By Jensen’s inequality, we can deduce that or N(T ) − Nd(T ) � T L 4πR log � 1 T � T κd � 1 − 1 2R log � 1 lim sup T →∞ T 2 � � � �fB � T 2 � �� 2 1 R �� − + it dt 2 L � � � �fB � �� 2 1 R �� − + it dt . 2 L + O(T ) All we need is to get an asymptotic formula for the mean square of fB. We have f(s) = (g + g0)ζ(s) + g1 � ′ H L H (s)ζ(s) + ζ′ � (s) = � � s � � log 2π 1 d �1 � −1 Q1 + ζ(s) + O |t| 2L L ds �� , where Q1(x) = g + g0 + g1x. Using the last two equations, integration by parts leads us to κd � 1 − 1 2R log � � � T � �� 2 1 � � 1 R �� T �V B − + it dt , 2 L lim sup T →∞ where V (s) = Q � − 1 � � d 1 ζ(s), and Q(x) = Q1 L ds 2 − x� = g + g0 + 1 2g1 − g1x. When we applied the Littlewood lemma in (4.4), we needed the condition Q(0) = 1. This yields Q(x) = 1 − g1x. In [2, p.10], Conrey made an optimal choice of this case. If we select g1 = 1.02, R = 1.2, θ = 4 7 References 2 , we have log c R = 0.598.... Therefore, we conclude that κd > 0.70. [1] B. C. Berndt, The number of zeros for ζ (k) (s), J. London Math. Soc.(2) 2 (1970), 577– 580 [2] J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. reine angew. Math. 399 (1989), 1–26 [3] J. B. Conrey, A. Ghosh, and S. M. Gonek, Simple zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 76 (1998), 497–522 [4] J. M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982), 219–288 8
[5] D. W. Farmer, Counting distinct zeros of the Riemann-zeta function, Electron. J. Combin. 2 (1995), Research Paper 1, approx. 5 pp. [6] N. Levinson, Gap and Density Theorems, Amer. Math. Soc. Colloq. Publ. Vol. 26, 1940. [7] N. Levinson, More than One Third of Zeros of Riemann’s Zeta-Function are on σ = 1/2, Adv. Math. 13 (1974), 383–436 [8] N. Levinson and H. L. Montgomery, Zeros of the derivatives of the Riemann zetafunction, Acta Math. 133 (1974), 49–65 [9] A. Selberg, On the zeros of Riemann’s zeta function. Skr. Norske Vid. Akad. Oslo, (1942) no. 10. [10] E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., revised by D. R. Heath-Brown, Oxford University Press, Oxford, 1986 9
Page 1 and 2: Zeros of the Derivatives of the Rie
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Zeros of the Derivatives of the Riemann Zeta-function ∗

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