How can I get the coefficient of a Dirichlet series?

asked 2015-04-15 18:10:04 +0200

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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.

Comments

It's not a power series, it's a Dirichlet series, see https://en.wikipedia.org/wiki/Dirichl...

rws ( 2015-04-16 09:01:41 +0200 )edit

I see. I edited the title. Thanks.

Semin ( 2015-04-16 15:11:17 +0200 )edit

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answered 2015-04-15 20:37:49 +0200

slelievre
17654 ●22 ●160 ●348 http://carva.org/samue...

See ask-sage question 10082 and trac ticket 16477.

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

rws ( 2015-04-16 09:00:18 +0200 )edit

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"??

Semin ( 2015-04-16 15:28:56 +0200 )edit

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]

rws ( 2015-04-16 18:23:52 +0200 )edit

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?

Semin ( 2015-04-17 06:47:11 +0200 )edit

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.

rws ( 2015-04-17 10:25:52 +0200 )edit

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