TITLE: Conjecture of Spira on the zeros of $\zeta^\prime(s)$
QUESTION [9 upvotes]: Let $N(T)$ be the number of complex zeros of $\zeta(s)$ with imaginary part between $0$ and $T$, and let $N_k(T)$ be the analogous counting function for the $k$th derivative $\zeta^{(k)}(s)$.  Based on numerical evidence for $T<100$, Spira conjectured in 1965 ("Zero free regions of $\zeta^{(k)}(s)$, J. London. Math. Soc.  v. 40 1965 pp. 677–682) that
$$
N(T)=N_k(T)+[T\log(2)/(2\pi)]\pm 1.
$$
Berndt later showed that
$$
N(T)=N_k(T)+T\log(2)/(2\pi)+O(\log(T)).
$$
Is Spira's original conjecture still open?  (I don't expect this is true; finding a counterexample will be a nice project for an undergraduate.)

REPLY [6 votes]: Shorokhodov gave an explicit counterexample to Spira's conjecture in "Pade approximates and numerical analysis of the Riemann zeta function", Computational Mathematics and Mathematical Physics, vol. 43 no. 9 (2003) pp. 1277-1298.  He computed, for $T=1420$, 1000 zeros of $\zeta(s)$, 844 zeros of $\zeta^\prime(s)$, and noted $1420\log 2/2\pi\approx 156.65$.