Zeros of derivatives

Much attention focuses on zeros of L-functions, but there are also some interesting questions about zeros of the derivatives of these functions.

In some cases we know more about zeros of the derivatives. For example, Levinson's method [58 #27837][ MR 84g:10070] can be used to show that for any $\epsilon>0$ there is an $N$ such that if $n\ge N$ then more than $(100-\epsilon)\%$ the zeros of $\xi^{(n)}(s)$ are on the critical line, where $\xi(s)$ is the Riemann $\xi$-function.

In other cases we know almost nothing. For example, nobody has even made a plausible conjecture about the distribution of the zeros of the derivatives of the Riemann $\zeta$-function.

See the articles on derivatives of $\xi(s)$ and derivatives of $\zeta(s)$.

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