Hello,
I am a bit puzzled by the following behaviour:
>
> sage: chi = DirichletGroup(1)[0]
> sage: chi.is_primitive()
> True
> sage: chi.is_trivial()
> True
> sage: chi(0)
> 1
> sage: chi(1)
> 1
> sage: chi(2)
> 1
> sage: chi.bernoulli(1)
> -1/2
>
> According to the definition given in the docstring of chi.bernoulli(), it
> should be 1/2, not -1/2. This also matches the convention used in
> Washington's "Cyclotomic Fields", and also what I find in the wikipedia
> article on generalised bernoulli numbers.
>
This should indeed be fixed (either the code or the documentation). We
also have
sage: chi = DirichletGroup(1)[0]
sage: chi.bernoulli(1, algorithm='definition', cache=False)
1/2
With the default algorithm='recurrence' option, the answer -1/2 is caused
by the fact that Sage uses the bernoulli function, which is often defined
to take the value -1/2 at 1 ("first Bernoulli numbers"):
sage: bernoulli(1)
-1/2
I'll open a ticket for this. The question is whether we want
chi.bernoulli(1) to return 1/2 (matching the definition of generalised
Bernoulli numbers) or -1/2 (matching the bernoulli function). I would
personally be in favour of 1/2, but maybe others here have a different
opinion.
Peter
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