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Defining Dirichlet series

asked 2013-05-02 07:45:54 -0500

kcrisman gravatar image

updated 2015-01-14 04:59:44 -0500

FrédéricC gravatar image

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)

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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answered 2014-02-09 01:50:02 -0500

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 :)

kcrisman gravatar imagekcrisman ( 2014-02-10 05:34:39 -0500 )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? :-)

KnS gravatar imageKnS ( 2014-05-31 08:58:29 -0500 )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!

kcrisman gravatar imagekcrisman ( 2014-06-02 02:47:03 -0500 )edit

For reference this is now #16477 at .

slelievre gravatar imageslelievre ( 2014-07-07 23:52:19 -0500 )edit

answered 2015-03-17 09:13:30 -0500

kcrisman gravatar image

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 failed since pexpect doesn't seem to like multiple arguments.

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Asked: 2013-05-02 07:45:54 -0500

Seen: 597 times

Last updated: Mar 17 '15