extend our analysis to the zeros of $\xi'(s)$, where $$\begin{aligned}
\xi(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma\bigg(\frac{s}{2}\bigg)\zeta(s).\end{aligned}$$ It is known that $\xi'(s)$ has only zeros in the critical strip $0<{{\rm Re}\,}s<1$ and that RH implies that all its zeros satisfy ${{\rm Re}\,}s=1/2$. Let $N_1(T)$ count the number of zeros $\rho_1=\beta_1+i\gamma_1$ of $\xi'(s)$ (with multiplicity) such that $0< \gamma_1 \leq T$. It is also known that $$\begin{aligned}
N_1(T) =  (1+o(1)) \frac{T}{2\pi} \log T.\end{aligned}$$ We can then similarly define the function $$N_{1}^{*}(T):=\displaystyle\sum_{0<\gamma_1\leq T}m_{\rho_1},$$ where $m_{\rho_1}$ is the multiplicity of the zero $\rho_1$.

\[thmn1star\] Assuming RH we have $$\begin{aligned}
    N_1^*(T)\leq (1.1175+o(1))N_1(T).
    \end{aligned}$$

To the best of our knowledge, this bound is the current best.

Defining the functions $N_{1,s}(T)$ and $N_{1,d}(T)$ (quantity of simple and distinct zeros respectively) for $\xi'(s)$ and using the inequalities $${N_{1,s}(T) \geq 2 N_1(T) - N_1^*(T)}$$ and $${N_{1,d}(T) \geq \frac{3}{2}N_1(T)-\frac{1}{2}N_1^*(T)},$$ that can be derived the same way as for $\zeta(s)$, we obtain the following corollary.

Assuming RH we have $$N_{1,s}(T)\geq (0.8825+o(1))N_1(T). $$ and $$N_{1,d}(T)\geq (0.9412+o(1))N_1(T). $$

The best previous result is due to Farmer, Gonek and Lee [@FGL], showing that more than $85.83\% $ of the zeros of $\xi'(s)$ are simple. To the best of our knowledge, our new bounds are the current best.

Derivation of 