How to treat (multiple) zeta values as real symbolic constants?

[Brent Pym]

For a real number s, the value zeta(s) of the Riemann zeta function is real, but SageMath does not seem to recognize this fact.  Here is an example (SageMath 7.5.1):

sage: conjugate(zeta(3))
conjugate(zeta(3))

One might hope to simply get zeta(3)as output.  Indeed, this is how the sin function behaves:

sage: conjugate(sin(3))
sin(3)

Is there some way to treat zeta values (and more generally, multiple zeta values) as real symbolic constants in SageMath or is this something that has yet to be implemented?  It seems to be related to (but a bit different from) Trac ticket https://trac.sagemath.org/ticket/18141.

[kcrisman]

On Tuesday, March 28, 2017 at 10:15:22 AM UTC-4, Brent Pym wrote:

For a real number s, the value zeta(s) of the Riemann zeta function is real, but SageMath does not seem to recognize this fact.  Here is an example (SageMath 7.5.1):

sage: conjugate(zeta(3))
conjugate(zeta(3))

One might hope to simply get zeta(3)as output.

One would have to hard-code that into Pynac, is my guess. 

[Ralf Stephan]

Yes, the elementary functions had this implemented only recently, and I'm happy to add more knowledge like that.

[Brent Pym]

Yes, the elementary functions had this implemented only recently, and I'm happy to add more knowledge like that.

Great, thanks a lot!

Do I understand correctly that multiple polylogs and multiple zeta values are implemented in GiNaC/Pynac but not yet wrapped in SageMath?

[Brent Pym]

Also, I realized that, more generally, conjugate passes through sin because of the holomorphicity:

sage: conjugate(sin(1+I))
sin(-I + 1)

Maybe this would be nice for other holomorphic functions, like zeta?  But perhaps that's how you had in mind to implement it anyways.

[Ralf Stephan]

On Thursday, March 30, 2017 at 9:08:24 AM UTC+2, Brent Pym wrote:

> Do I understand correctly that multiple polylogs and multiple zeta values are implemented in GiNaC/Pynac but not yet wrapped in SageMath?

They are in GiNaC but the code in Pynac is degenerated for various reasons, eg the numerics use CLN which is not in Sage.

See also https://trac.sagemath.org/ticket/10050

[Ralf Stephan]

I also opened https://trac.sagemath.org/ticket/22713

[Brent Pym]

Thank you very much!  By opening these tickets, you have certainly answered my questions.  Is the etiquette to mark this thread as complete now, or wait until the tickets have been closed?

On Thursday, 30 March 2017 08:58:07 UTC+1, Ralf Stephan wrote:

I also opened https://trac.sagemath.org/ticket/22713

[Ralf Stephan]

On Thursday, March 30, 2017 at 10:38:49 AM UTC+2, Brent Pym wrote:

Thank you very much!  By opening these tickets, you have certainly answered my questions.  Is the etiquette to mark this thread as complete now, or wait until the tickets have been closed?

Threads in sage-devel or -support are mostly open-ended. Developers use trac tickets as todo list.

[Brent Pym]

Okay, thanks very much for all your prompt and helpful replies!

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