I'm not opposed in principle to the new definition. For example, a
nice feature of having B_1 = +1/2 is that it agrees with the most
common definition of generalized Bernoulli numbers when specialized to
the trivial character.
However, this would be inconsistent with Pari/GP, Magma, Maple, Mathematica, etc.,
not to mention several decades of mathematical literature. It is
unlikely that all those other systems will change their definition, and
having two conventions around is arguably worse than having a single
convention that is slightly less elegant. It's maybe not really a big deal
since most applications only use the even-index Bernoulli
numbers, but I'm sure this will break *some* user code.
Probably this should be brought up on sage-devel?
It's not enough to change the libraries if you want to get this into
Sage; if FLINT and Arb change their definitions but the Sage developers
decide that they don't like it, they will just treat the
new behavior as a bug and add a special case in the wrapper to return
B_1 = -1/2.
Fredrik