Riemann and Hurwitz Zeta functions for complex and real intervals

[cheuberg]

    sage: zeta(RIF(3))
    1.20205690315959
    sage: _.parent()
    Complex Field with 53 bits of precision
    sage: zeta(CIF(2, 3))
    0.798021985146276 - 0.113744308052939*I
    sage: _.parent()
    Complex Field with 53 bits of precision

(see #19993 comment:10).

Fix this by conversion to arb, computation of zeta there, and conversion back.
(mpfi does not seem to have a zeta function).

CC: @dkrenn @sagetrac-skropf @kcrisman @robertwb @mezzarobba

Component: numerical

Keywords: complex interval, CIF, Riemann zeta function

Author: Clemens Heuberger

Branch/Commit: 7a590fb

Reviewer: Fredrik Johansson

Issue created by migration from https://trac.sagemath.org/ticket/16782

[cheuberg]

comment:1

I do have a TeX file containing additional (boring) computations documenting parts of the code, but I am not sure where to put it, see my post on sage-devel.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

a947cfc

Trac #16782: Fix trivial typo in docstring

[sagetrac-git]

Changed commit from 315df6e to a947cfc

[sagetrac-git]

Changed commit from a947cfc to 67bee31

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

67bee31

Trac #16782: Fix function name in verbose output.

[fredrik-johansson]

comment:4

Riemann zeta for complex intervals could also be implemented using the Arb library (#16747).

[sagetrac-git]

Changed commit from 67bee31 to 4608c67

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4608c67

Trac #16782: Add example and TODO concerning negative real values of s

[cheuberg]

comment:6

Replying to @fredrik-johansson:

Riemann zeta for complex intervals could also be implemented using the Arb library (#16747).

Thank you for pointing that out; once #16747 is included and there are conversions between CIF and arb, that would certainly be the better option.

[cheuberg]

Changed branch from u/cheuberg/rings/zeta-CIF to none

[cheuberg]

comment:8

Deleted the branch as it was not the way to proceed.

[cheuberg]

Changed commit from 4608c67 to none

[cheuberg]

Branch: u/cheuberg/rifcif-zeta

[cheuberg]

New commits:

7a590fb

Trac #16782: implement zeta function for RIF and CIF

[cheuberg]

Commit: 7a590fb

[fredrik-johansson]

Reviewer: Fredrik Johansson

[fredrik-johansson]

comment:13

This looks good to me.

[vbraun]

Changed branch from u/cheuberg/rifcif-zeta to 7a590fb