ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 17 Mar 2015 09:13:30 -0500Defining 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.Thu, 02 May 2013 07:45:54 -0500http://ask.sagemath.org/question/10082/defining-dirichlet-series/Answer by slelievre for <p>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 <a href="http://en.wikipedia.org/wiki/Dirichlet_series">Dirichlet series</a>. 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).</p>
<p>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</p>
<pre><code>L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1)
L.init_coeffs('moebius(k)')
</code></pre>
<p>and the documentation for <code>Dokchitser</code> seems to indicate this might be valid. But the numbers I get are wrong.</p>
<p>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? </p>
<p>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.</p>
http://ask.sagemath.org/question/10082/defining-dirichlet-series/?answer=16026#post-id-16026[John Hanke](http://www.wordpress.jonhanke.com/) has a [Math software](http://www.wordpress.jonhanke.com/?page_id=128) page where item number 6 is [Formal Dirichlet series in Sage](http://www.wordpress.jonhanke.com/Software/Sage__Dirichlet_series/Dirichlet_series.sage).
Sun, 09 Feb 2014 01:50:02 -0600http://ask.sagemath.org/question/10082/defining-dirichlet-series/?answer=16026#post-id-16026Comment by kcrisman for <p><a href="http://www.wordpress.jonhanke.com/">John Hanke</a> has a <a href="http://www.wordpress.jonhanke.com/?page_id=128">Math software</a> page where item number 6 is <a href="http://www.wordpress.jonhanke.com/Software/Sage__Dirichlet_series/Dirichlet_series.sage">Formal Dirichlet series in Sage</a>.</p>
http://ask.sagemath.org/question/10082/defining-dirichlet-series/?comment=16168#post-id-16168No, I haven't had time or opportunity. It's certainly not in Sage yet, and I don't think I ever opened a ticket. Feel free to!Mon, 02 Jun 2014 02:47:03 -0500http://ask.sagemath.org/question/10082/defining-dirichlet-series/?comment=16168#post-id-16168Comment by KnS for <p><a href="http://www.wordpress.jonhanke.com/">John Hanke</a> has a <a href="http://www.wordpress.jonhanke.com/?page_id=128">Math software</a> page where item number 6 is <a href="http://www.wordpress.jonhanke.com/Software/Sage__Dirichlet_series/Dirichlet_series.sage">Formal Dirichlet series in Sage</a>.</p>
http://ask.sagemath.org/question/10082/defining-dirichlet-series/?comment=16169#post-id-16169Hey @KCrisman, Did you try this out? Is there a ticket somewhere on Sage Trac pointing to this? Or is it already in Sage? :-)Sat, 31 May 2014 08:58:29 -0500http://ask.sagemath.org/question/10082/defining-dirichlet-series/?comment=16169#post-id-16169Comment by kcrisman for <p><a href="http://www.wordpress.jonhanke.com/">John Hanke</a> has a <a href="http://www.wordpress.jonhanke.com/?page_id=128">Math software</a> page where item number 6 is <a href="http://www.wordpress.jonhanke.com/Software/Sage__Dirichlet_series/Dirichlet_series.sage">Formal Dirichlet series in Sage</a>.</p>
http://ask.sagemath.org/question/10082/defining-dirichlet-series/?comment=16295#post-id-16295That is awesome! Why did this never make it into Sage? If it works right I'll accept this for sure :)Mon, 10 Feb 2014 05:34:39 -0600http://ask.sagemath.org/question/10082/defining-dirichlet-series/?comment=16295#post-id-16295Comment by slelievre for <p><a href="http://www.wordpress.jonhanke.com/">John Hanke</a> has a <a href="http://www.wordpress.jonhanke.com/?page_id=128">Math software</a> page where item number 6 is <a href="http://www.wordpress.jonhanke.com/Software/Sage__Dirichlet_series/Dirichlet_series.sage">Formal Dirichlet series in Sage</a>.</p>
http://ask.sagemath.org/question/10082/defining-dirichlet-series/?comment=23258#post-id-23258For reference this is now #16477 at http://trac.sagemath.org/ticket/16477 .Mon, 07 Jul 2014 23:52:19 -0500http://ask.sagemath.org/question/10082/defining-dirichlet-series/?comment=23258#post-id-23258Answer by kcrisman for <p>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 <a href="http://en.wikipedia.org/wiki/Dirichlet_series">Dirichlet series</a>. 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).</p>
<p>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</p>
<pre><code>L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1)
L.init_coeffs('moebius(k)')
</code></pre>
<p>and the documentation for <code>Dokchitser</code> seems to indicate this might be valid. But the numbers I get are wrong.</p>
<p>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? </p>
<p>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.</p>
http://ask.sagemath.org/question/10082/defining-dirichlet-series/?answer=26219#post-id-26219Apparently Pari also has this functionality, or something like it. See [this math.SX question](http://math.stackexchange.com/questions/1193708/question-about-direuler-command-in-pari-gp) for details. Unfortunately all my attempts to do this with `sage.gp` failed since pexpect doesn't seem to like multiple arguments.Tue, 17 Mar 2015 09:13:30 -0500http://ask.sagemath.org/question/10082/defining-dirichlet-series/?answer=26219#post-id-26219