Zeros of 𝜁^{’}(𝑠) in the critical strip

Spira, Robert

doi:10.1090/S0002-9939-1972-0296035-5

Zeros of $\zeta ^{’} (s)$ in the critical strip
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Proc. Amer. Math. Soc. 35 (1972), 59-60 Request permission

Abstract:

It is shown that the abscissa of convergence for the Dirichlet series ${( - 1)^k}{(1 - {2^{1 - s}})^{k + 1}}{\zeta ^{(k)}}(s)$ is zero, where $\zeta (s)$ is the Riemann zeta function. This implies the existence of infinitely many zeros of $\zeta ’(s)$ in the critical strip.

References

Robert Spira, Zero-free regions of $\zeta ^{(k)}(s)$, J. London Math. Soc. 40 (1965), 677–682. MR 181621, DOI 10.1112/jlms/s1-40.1.677
Robert Spira, Another zero-free region for $\zeta ^{(k)}\,(s)$, Proc. Amer. Math. Soc. 26 (1970), 246–247. MR 263754, DOI 10.1090/S0002-9939-1970-0263754-4
Robert Spira, On the Riemann zeta function, J. London Math. Soc. 44 (1969), 325–328. MR 260684, DOI 10.1112/jlms/s1-44.1.325

Zeros of

and the Riemann hypothesis

Bruce C. Berndt, The number of zeros for $\zeta ^{(k)}\,(s)$, J. London Math. Soc. (2) 2 (1970), 577–580. MR 266874, DOI 10.1112/jlms/2.Part_{4}.577

Ein Satz über Dirichletsche Reihen mit Anwendung auf die

Funktion und die L-Funktionen

37

Theory of functions