ENH: special: change zeta signature to zeta(x, q=1)

[pv]

Make the second argument of zeta(x,q) default to q=1. This makes the one-argument zeta(x) be the Riemann zeta function if the second argument is not given.

Previously, you had to write zeta(x,1) which is somewhat annoying since the Riemann zeta function is more often needed than the Hurwitz one.

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[person142]

I've wondered whether it was worth introducing a new function called maybe rzeta for the Riemann zeta function, deprecating zeta and moving its functionality to a function called maybe hzeta. The reasoning is:

• It's better to have a custom implementation of the Riemann zeta function as it will have better accuracy than a general algorithm for the Hurwitz zeta function. (The reflection formula is much nicer and extra care can be taken around the critical strip.)
• You could just link zeta(x, 1) to the custom implementation, but I think it's generally nice for the regions on which functions work to be (roughly) continuous in the arguments.

Of course all this is moot without implementations of Riemann/Hurwitz zeta for complex arguments, which I'm still working on...

[pv]

The continuous alternative would be to have something like zeta(x, q=None) to indicate the non-Hurwitz implementation. I'd like to have the name "zeta" reserved for the Riemann zeta function, because it's the most common one.

[pv]

Updated --- maybe this is a better way to do the riemann/hurwitz switch

[person142]

This seems like a nice solution to me. Perhaps worth adding a note in the documentation that Riemann zeta is Hurwitz zeta with q = 1?

Also, strictly speaking the domain in the documentation is incorrect; things like zeta(n, -x) (n > 1 an integer, x > 0 not an integer) work... however this domain is some kind of madness we might want to pretend doesn't exist.

[ev-br]

Maybe even mention this in the release notes. Regardless, +1 and LGTM.

[endolith]

@pv So what's the rationale for def zeta(x, q=None, out=None) instead of the more informative def zeta(x, q=1, out=None)?

[pv]

See the discussion above.

[endolith]

@pv I did, but I don't see a reason for why it should default to q=None instead of q=1

[pv]

Because we want to leave room to use different algorithms for hurwitz and plain zeta.
For example, complex values etc

[endolith]

Support complex values and using different algorithms is great, but what does that have to do with the q argument? It can switch to a different algorithm for q==1 just as easily as q==None

[endolith]

Ohh, in case the algorithms produce slightly different results, you don't want a discontinuity when sweeping across q? If that's why, I think it would make sense to have 2 different functions

[person142]

Ohh, in case the algorithms produce slightly different results, you don't want a discontinuity when sweeping across q

Yes, that's the reasoning. Since Riemann zeta is so well-studied there are superior algorithms. Two different functions makes sense, but backwards compatibility.