# 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. 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.

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Apparently Pari also has this functionality, or something like it. See this math.SX question for details. Unfortunately all my attempts to do this with sage.gp failed since pexpect doesn't seem to like multiple arguments.

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John Hanke has a Math software page where item number 6 is Formal Dirichlet series in Sage.

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That is awesome! Why did this never make it into Sage? If it works right I'll accept this for sure :)

( 2014-02-10 12:34:39 +0200 )edit

Hey @kcrisman, Did you try this out? Is there a ticket somewhere on Sage Trac pointing to this? Or is it already in Sage? :-)

( 2014-05-31 15:58:29 +0200 )edit

No, 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!

( 2014-06-02 09:47:03 +0200 )edit
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For reference this is now #16477 at http://trac.sagemath.org/ticket/16477 .

( 2014-07-08 06:52:19 +0200 )edit