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Dedekind Zeta function of cyclotomic field wrongly evaluating to zero on -1?

asked 2021-02-06 23:44:49 +0200

tzeentch gravatar image

updated 2021-07-15 20:20:56 +0200

FrédéricC gravatar image

Let $K := \mathbb{Q}(\zeta)$ be the pth cyclotomic extension of $\mathbb{Q}$. I would like to verify the results of a paper which states the quotient of their Dedekind zeta functions have particular values. Below Z is the Riemann zeta function (Dedekind zeta function of $\mathbb{Q}$).

x = var('x')
K = NumberField(x**2 + x + 1,'a')
L = K.zeta_function(algorithm='gp')
Z = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')

The expected values are nonzero! For example. (L/Z)(-1) is expected to be 1.333333333 (i.e. 4/3).

L(-1) returns 0.000000000000000, as does L(-1)/Z(-1).

Z(-1) returns -0.0833333333333333. L/Z returns a type error, as does L(x)/Z(x).

Here is my first question: Have I incorrectly implemented the Dedekind Zeta function of a cyclotomic number field? Why is L(-1) = 0?

Here is my second question: How do I implement the evaluation of the L-series after I've taken their quotient? That is, A = L/Z, A(-1); instead of L(-1)/Z(-1).

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answered 2021-02-07 09:30:12 +0200

FrédéricC gravatar image

updated 2021-02-07 11:27:04 +0200

Do not use Dokchitser implementation, but always the pari implementation.

sage: K = NumberField(x**2 + x + 1,'a')                                         
sage: Q = NumberField(x-1,'y')          # using QQ should work, this is a workaround
sage: KL = K.zeta_function()                                                    
sage: QL = Q.zeta_function()                                                    
sage: KL(-1)                                                                    
sage: QL(-1)                                                                    

so the quotient will be zero. You can use taylor series as follows

sage: KL.taylor_series(-1,4)                                                    
0.000000000000000 - 0.0269221622682875*z - 0.0573141973539488*z^2 - 0.0443122899350116*z^3 + O(z^4)

We also have L functions for Dirichlet characters.

sage: D = DirichletGroup(4)                                                     
sage: chi = D.gen(0)                                                            
sage: chi.lfunction()                                                           
PARI L-function associated to Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1
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So the value of KL(-1) is indeed zero. This is very troubling! Thank you.

tzeentch gravatar imagetzeentch ( 2021-02-07 20:00:38 +0200 )edit

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Asked: 2021-02-06 23:44:49 +0200

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Last updated: Jul 15 '21