Conjecture of Spira on the zeros of $\zeta^\prime(s)$

Question

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.)

Answer 1

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.

Answer 2

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 $1420\log 2/2\pi\approx 156.65$.