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.)



Conjecture of Spira on the zeros of $\zeta^\prime(s)$
I'm looking at the review, by Haseo Ki, of Hirotaka Akatsuka, Conditional estimates for error terms related to the distribution of zeros of $\zeta'(s)$, J. Number Theory 132 (2012), no. 10, 2242–2257, MR2944752. It says, 


Assuming the Riemann hypothesis, the author shows $$N(T)=N\_1(T)+{T\log2\over2\pi}+O\left({\log T\over\sqrt{\log\log T}}\right)$$ and comments that there is a barrier to further improvement of this.


