4
$\begingroup$

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 $\frac{1}{2}<\Re(z)<1$

Thanks

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.