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

kcrisman gravatar image

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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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.