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A note on $\zeta ''(s)$ and $\zeta '''(s)$

Author(s): C. Yalçin Yildirim
Journal: Proc. Amer. Math. Soc. 124 (1996), 2311-2314.
MSC (1991): Primary 11M26
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Abstract: There is only one pair of non-real zeros of $\zeta ''(s)$ , and of $\zeta '''(s)$ , in the left half-plane. The Riemann Hypothesis implies that $\zeta ''(s)$ and $\zeta '''(s)$ have no zeros in the strip $0 \leq \Re s < \frac 12 %$ .

References:

1.: T. M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Math. Comp. 44 (1985), 223-232. MR 86c:11063
2.: N. Levinson and H. L. Montgomery, Zeros of derivatives of the Riemann zeta-function, Acta Math. 133 (1974), 49-65. MR 54:5135
3.: A. Speiser, Geometrisches zur Riemannschen zetafunktion, Math. Ann. 110 (1934), 514-521.
4.: R. Spira, Zero-free regions of $\zeta ^{(k)}(s)$ , J. London Math. Soc. 40 (1965), 677-682. MR 31:5849
5.: E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., Oxford, 1986. MR 88c:11049
6.: C. Y. Yildirim, A note on $\zeta ''(s)$ and $\zeta '''(s)$ , detailed version, manuscript available by e-mail.


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