This is now http://trac.sagemath.org/ticket/17586 .
Op maandag 5 januari 2015 15:55:33 UTC+1 schreef Peter Bruin:
>
> 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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