ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 06 Feb 2021 23:44:49 +0100Dedekind Zeta function of cyclotomic field wrongly evaluating to zero on -1?https://ask.sagemath.org/question/55607/dedekind-zeta-function-of-cyclotomic-field-wrongly-evaluating-to-zero-on-1/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)`. masseygirlSat, 06 Feb 2021 23:44:49 +0100https://ask.sagemath.org/question/55607/Defining Dirichlet serieshttps://ask.sagemath.org/question/10082/defining-dirichlet-series/In basic analytic number theory, before one really starts talking about crazy L-functions of elliptic curves and the like, you can introduce so-called [Dirichlet series](http://en.wikipedia.org/wiki/Dirichlet_series). It is especially nice because the concepts really are accessible to anyone who has had a good calculus course and knows some elementary number theory (you don't have to talk about complex numbers, at first).
I have wanted to use these in Sage for a long time, but never seem to quite find the right command. For example, for the series defined by Moebius $\mu$, I want to use
L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1)
L.init_coeffs('moebius(k)')
and the documentation for `Dokchitser` seems to indicate this might be valid. But the numbers I get are wrong.
Since I don't really know that much about L-functions in general, it's possible that the $\mu$ function's series has a different conductor or weight or something. But it wasn't easy to find any connections to this more general theory. Can someone help?
Bonus: if we can wrap this (or some other Sage) functionality to provide Dirichlet series for all kinds of things, including the Dirichlet L-functions for showing off the theorem on primes in an arithmetic progression and so forth, it would make a nice patch.kcrismanThu, 02 May 2013 14:45:54 +0200https://ask.sagemath.org/question/10082/