Zeros of \(\zeta''(s)\) \& \(\zeta'''(s)\) in \(\sigma<\frac 12\) (Q2703144)

It is shown that \(\zeta''(s)\) and \(\zeta'''(s)\) each have only one conjugate pair of non-real zeros in the half-plane \(\text{Re}\,s<0\). Assuming the Riemann Hypothesis, it is shown that neither function vanishes in the strip \(0\leq\text{Re}\,s<1/2\). The proofs use partial fraction decompositions of \(\zeta''(s)/ \zeta'(s)\) and \(\zeta'''(s)/\zeta''(s)\), and require a certain amount of computation. The statements of the results have already appeared in the author's paper [Proc. Am. Math. Soc. 124, 2311--2314 (1996; Zbl 0863.11057)].