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Related to this and this.

Let $K$ be the number field with the degree 24 defining polynomial

 x^24 + 3*x^23 - 2*x^22 - 43*x^21 + 81*x^20 + 1579*x^19 + 2434*x^18 - 5192*x^17 + 4678*x^16 - 41425*x^15 + 423527*x^14 + 1352722*x^13 + 5199537*x^12 - 13364304*x^11 - 138065100*x^10 + 228783352*x^9 + 1254448448*x^8 - 3179566016*x^7 + 4205123840*x^6 + 139822208*x^5 - 31439415040*x^4 + 28607489536*x^3 + 330701977600*x^2 - 807251576832*x + 635017424896

and $\zeta_K$ be the zeta function of $K$.

According to an answer in the second question and Magma:

$\zeta_K=\zeta\cdot L(\omega)\cdot L(\bar\omega)\cdot L(\tau_2)^2\cdot L(\tau_2\omega)^2\cdot L(\tau_2\bar\omega)^2\cdot L(\kappa_3)^3$

Zero of each factor of multiplicity greater than one is a double or triple zero of $\zeta_K$.

Q1 Does the infinitely many double zeros imply some properties of $K$?

Q2 Why should one care about the simplicity of the zeros of Riemann zeta function in light of these double zeros?

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    $\begingroup$ For Q2, usually one considers primitive $L$-functions, and the only Dedekind $\zeta$-function that is primitive is that for the rationals. So the fact that $\zeta_K$ for general $K$ can have higher order zeros (due to primitive factors occurring with high multiplicity) is not so relevant. $\endgroup$
    – user334725
    Commented Sep 22, 2021 at 15:26
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    $\begingroup$ Those double zeros are artificial, because they are coming from an $L$-function that appears inside the zeta-function of $K$ with multiplicity greater than $1$. It's sort of like asking if the repeated prime factors in a perfect square $n^2$ for $n > 1$ could imply some property of prime numbers. What is interesting are higher-order (meaning not of order $1$) nontrivial zeros for an $L$-function that does not (or is at least not expected to) decompose into other $L$-functions; such phenomena do happen in the setting of the Birch and Swinnerton-Dyer conjecture. $\endgroup$
    – KConrad
    Commented Sep 23, 2021 at 0:24
  • $\begingroup$ @KConrad In your example with $n^2$ I have the properties "never prime" and "always squareful", which would be an answer to the question. $\endgroup$
    – joro
    Commented Sep 23, 2021 at 16:48

1 Answer 1

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A1: The existence of zeros of high multiplicity for $\zeta_K(s)$ is connected to whether the analogue of Mertens's conjecture for $K$ is true. See here.

A2: The simplicity of the zeros of $\zeta(s)$, or any other irreducible $L$-function, is of great interest. For example, the simplicity of the zeros of $\zeta(s)$ is a hypothesis (along with the Riemann Hypothesis) in a theorem due to Rubinstein and Sarnak whose conclusion is that the logarithmic density of real numbers $x>0$ such that $\mathrm{Li}(x)>\pi(x)$ is

$0.99999973\ldots$

Similar bias questions can be posed with other $L$-functions. Weaker (but still quite interesting) conclusions can be drawn using weaker hypotheses (such as bounded multiplicity of zeros instead of simplicity). See here and here, for example. But it seems to be the case that one needs the full simplicity hypothesis to achieve the above logarithmic density.

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  • $\begingroup$ By irreducible L-function, do you mean primitive element of the Selberg class? $\endgroup$
    – Sylvain JULIEN
    Commented Sep 22, 2021 at 21:58
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    $\begingroup$ @SylvainJULIEN I mean an L-function that does not factor as a product of L-functions of smaller degree. $\endgroup$
    – 2734364041
    Commented Sep 22, 2021 at 23:12
  • $\begingroup$ Ok, so that's the same notion. $\endgroup$
    – Sylvain JULIEN
    Commented Sep 23, 2021 at 9:23

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