ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 05 Dec 2018 11:17:59 -0600Understanding Output in SageMath Regarding Dirichlet Charactershttp://ask.sagemath.org/question/44593/understanding-output-in-sagemath-regarding-dirichlet-characters/ p=7
G = DirichletGroup(p); G
m=3; n=ZZ((p-1)/m); print m,n
c=G[1]
c1=c^n;c1
The output is:
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1
Can anyone explain what zeta6 is? Is this the Riemann-Zeta function? Is this the whole group of units? Is there a relation to the Eisenstein primes? I'm still a bit weak in this material and am having trouble grasping some of these sage outputs. Thank you in advance!NicklovnWed, 05 Dec 2018 11:17:59 -0600http://ask.sagemath.org/question/44593/How can I get the coefficient of a Dirichlet series?http://ask.sagemath.org/question/26537/how-can-i-get-the-coefficient-of-a-dirichlet-series/Hello.
Let's see this example.
g(x)=(1-3^(-x))*(f(x))^2
where f(x) is the Riemann zeta function and x is complex variable.
If Re(s) is sufficiently large then g(x) is converges.
We only view this g(x) as a formal Dirichlet series.
What I want is coefficients.
The Riemann zeta function is rewritten by
f(x)=1+2^(-x)+3^(-x)+ ...
We can also rewrite the g(x) by the sum of `a_n * n^(-x)`.
g(x)=sum{a_n * n^(-x) | n=1,2,...}
For given `n`, how can I get `a_n` ??
Is there any helpful sage command ??
Thanks.SeminWed, 15 Apr 2015 11:10:04 -0500http://ask.sagemath.org/question/26537/Defining Dirichlet serieshttp://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 07:45:54 -0500http://ask.sagemath.org/question/10082/