Define the Airy zeta function for $n=2,3,\dots$, by $$ Z(n) := \sum_r \frac{1}{r^n}. $$ where the sum is over the zeros $r$ of the Airy function $\operatorname{Ai}$.

In Mathematica the $\operatorname{Ai}$ function is implemented as AiryAi, and the zeros of this function is implemented as AiryAiZero.

I have tried to calculate the values of $Z$ as the following: Z[n_] := Sum[1/AiryAiZero[k]^n, {k, 1, Infinity}], and then N[Z[2]] for example.

Sadly in Mathematica 9.0 it gives $0.499$, however the correct result is $0.531457$. For larger $n$ values $Z$ is correct, however for only few digits, even if I modify $MaxExtraPrecision.

The Airy zeta function Mathworld page gives a closed-form of $Z$, but then you have to implement $n$th derivatives of different Airy-related functions.

How to implement $Z$ numerical efficient?

Define the Airy zeta function for $n=2,3,\dots$, by $$ Z(n) := \sum_r \frac{1}{r^n}. $$ where the sum is over the zeros $r$ of the Airy function $\operatorname{Ai}$.

In Mathematica the $\operatorname{Ai}$ function is implemented as AiryAi, and the zeros of this function is implemented as AiryAiZero.

I have tried to calculate the values of $Z$ using the following: Z[n_] := Sum[1/AiryAiZero[k]^n, {k, 1, Infinity}], and then N[Z[2]] for example.

Sadly, in Mathematica 9.0 it gives $0.499$; however, the correct result is $0.531457$. For larger $n$ values $Z$ is correct; however, for only a few digits, even if I modify $MaxExtraPrecision.

The Airy zeta function MathWorld page gives a closed-form of $Z$, but then you have to implement $n$th derivatives of different Airy-related functions.

How do I implement an efficient $Z$ function?

Define the Airy zeta function for $n=2,3,\dots$, by $$ Z(n) := \sum_r \frac{1}{r^n}. $$ where the sum is over the zeros $r$ of the Airy function $\operatorname{Ai}$.

In Mathematica the $\operatorname{Ai}$ function is implamented as AiryAi, and the zeros of this function is implamanted as AiryAiZero.

I have tried to calculate the values of $Z$ as the following: Z[n_] := Sum[1/AiryAiZero[k]^n, {k, 1, Infinity}], and then N[Z[2]] for example.

Sadly in Mathematica 9.0 it gives $0.499$, however the correct result is $0.531457$. For larger $n$ values $Z$ is correct, however only gives for few correct digits, even if I modify $MaxExtraPrecision.

The Airy zeta function Mathworld page gives a closed-form of $Z$, but then you have to implement $n$th derivatives of different Airy-related functions.

How to implement $Z$ numerical efficient?

Define the Airy zeta function for $n=2,3,\dots$, by $$ Z(n) := \sum_r \frac{1}{r^n}. $$ where the sum is over the zeros $r$ of the Airy function $\operatorname{Ai}$.

In Mathematica the $\operatorname{Ai}$ function is implemented as AiryAi, and the zeros of this function is implemented as AiryAiZero.

I have tried to calculate the values of $Z$ as the following: Z[n_] := Sum[1/AiryAiZero[k]^n, {k, 1, Infinity}], and then N[Z[2]] for example.

Sadly in Mathematica 9.0 it gives $0.499$, however the correct result is $0.531457$. For larger $n$ values $Z$ is correct, however for only few digits, even if I modify $MaxExtraPrecision.

The Airy zeta function Mathworld page gives a closed-form of $Z$, but then you have to implement $n$th derivatives of different Airy-related functions.

How to implement $Z$ numerical efficient?