Let $$ \eta^{(d)}(z) = \sum_{n=1}^\infty \dfrac {(-1)^d(-1)^{n-1}\ln(n)^d} {n^z} $$ be the derivative of Dirichlet Eta function of order $d$.

Does it exist any known or not known zero of $\eta^{(d)}(z)$ such that $d \geq 1$ and $\frac{1}{2}<\Re(z)<1$



Not an answer, but relevant: Bohr and Landau showed in [1] that if a Dirichlet series converges for $\sigma>0$, then $N(\alpha,T)$, the number of zeros for $\sigma>\alpha$ and $0<t<T$, is $O(T)$ for $\alpha>1/2$.

Update ($d=1$ case): The Mathematica command

FindRoot[2^(1 - s) Log[2] Zeta[s] + (1 - 2^(1 - s)) Zeta'[s], {s,1. + 95. I}]

returns {s -> 0.926336 + 95.3143 I}

[1] Ein Satz über Dirichletsche Reihen mit Anwendung auf die $\zeta$ Funktion und die $L$-Funktionen, Rend. Circ. Mat. Palermo 37 (1914), 269-272.

  • $\begingroup$ Thanks @Stopple for your suggestion, so we don't know if $\eta^{(d)}(z)=0$ when $d \geq 1$ and $\frac{1}{2} < \Re(z) < 1$ ? $\endgroup$ – Matey Math Feb 10 at 23:58
  • $\begingroup$ thanks @Stopple for your example $\endgroup$ – Matey Math Feb 13 at 13:41

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