How can I get the coefficient of a Dirichlet series?

asked 2015-04-15 11:10:04 -0500 This post is a wiki. Anyone with karma >750 is welcome to improve it.

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.

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The question will be resolved as soon as the ticket is reviewed. What coincidence...

That's awesome. Very useful information. Thanks. But I'm not sure how to use the commands in the "Dirichlet_series.sage" in order to (1-3(-s))*zeta__series(n)^2. How to control "s"??

Your x is usually named s (just a convention with complex variables), and to get the coefficients you would issue the commands (I have done it using a development version):

sage: s=var('s')
sage: dirichlet_series((1-3^(-s))*zeta(s)^2).list(20)
[1, 2, 3, 3, 2, 6, 2, 4, 6, 4, 2, 9, 2, 4, 6, 5, 2, 12, 2, 6]

I found "dirichlet_series.py". It contains "Copyright (C) 2015 Ralf Stephan". Maybe that's the development version you said. But I have some problem.

sage: s=var('s')
sage: dirichlet_series(zeta(s))
<repr(<sage.all_cmdline.DirichletSeries at 0x115ae6490>) failed: AttributeError: 'sage.libs.pari.gen.gen' object has no attribute 'dirmul'>
sage: dirichlet_series(1)
<repr(<sage.all_cmdline.DirichletSeries at 0x115ae68d0>) failed: AttributeError: 'sage.libs.pari.gen.gen' object has no attribute 'dirmul'>

Hmm.. It doesn't work. How to solve that problem?

Wait until release. You see, if you do not know how to deal with a possible bug in a development branch on trac, then you are not supposed to use it, anyway.