#16477: implement Dirichlet series
-------------------------------------+-------------------------------------
Reporter: rws | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-wishlist
Component: number theory | Resolution:
Keywords: moebius, zeta, | Merged in:
sigma, euler_phi, euler | Reviewers:
Authors: Jonathan Hanke, | Work issues: use pari, g.f. input
Ralf Stephan | Commit:
Report Upstream: N/A | 949082ca407de7df7ae2ce31ecfad4f5d21f3ffa
Branch: public/dirichlet- | Stopgaps:
series |
Dependencies: #18038, #18041 |
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Comment (by jj):
rws:
I'm in particular hoping for an exact implementation of
arithmetic/multiplicative functions:
{{{
id = ArithmeticFunctions().id()
sigma0 = ArithmeticFunction(lambda n: sigma0(n))
id*id == sigma0 // Dirichlet convolution
}}}
...
{{{
f = DirichletSeries(id)*DirichletSeries(id)
f == DirichletSeries(sigma0)
f[91234154]
f // nice representations:
=> sum_{n in Z} (sum_{d|n} 1)
or: sum_{n in Z} sigma0(n)
}}}
...
Those certainly form a nice subset of possible coefficient functions /
dirichlet series.
More generally I was hoping for the possibility to specify Dirichlet
series /
coefficient functions exactly, not only up to some precision.
E.g. (bad example) DirichletSeries(lambda n: sin(n))
or e.g. by specifying an (exact) generating series as e.g. a rational
function
Calculating the n'th coefficient from these to get a DirichletSeries as
now
is at least rather simple.
Other ideas: exact verification of identities (not only up to precision),
nice representations/output of coefficient functions after operations
(hard),
support functions from ideals (TODO: figure out the correct category for
the domain
of coefficient functions).
I am aware that this is not a small task and might not be easily possible
in python.
But I hope you get the idea of what I had in mind...
--
Ticket URL: <http://trac.sagemath.org/ticket/16477#comment:24>
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