Opened 5 years ago
Last modified 4 years ago
#21899 new enhancement
Document Stieltjes constants
Reported by: | cheuberg | Owned by: | |
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Priority: | minor | Milestone: | sage-7.5 |
Component: | symbolics | Keywords: | |
Cc: | behackl | Merged in: | |
Authors: | Reviewers: | ||
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | #21855 | Stopgaps: |
Description (last modified by )
According to https://en.wikipedia.org/wiki/Stieltjes_constants as well as to http://dlmf.nist.gov/25.2.E4, the Laurent series expansion of the Riemann zeta function around s=1 is
zeta(s) = 1/(s-1) + gamma_0 - gamma_1 (s-1) + O((s-1)^2).
However, sage says
sage: zeta(s).series(s==1, 2) 1*(s - 1)^(-1) + (euler_gamma) + (-1/2*stieltjes(1))*(s - 1) + Order((s - 1)^2)
(note the denominator 2 in the coefficient of (s-1)).
Numerically,
sage: stieltjes(1.) -0.0728158454836767
and Wikipedia states that
gamma_1 = −0.0728158454836767248605863758749013191377363383 ...
It seems that there is an incorrect factor in sage (this is now fixed and doctested).
When fixing this, the documentation of stieltjes should be improved to actually contain a definition of the stieltjes constants instead of simply stating that "The Stieltjes constants are used in the series expansions of zeta(s)."
Change History (5)
comment:1 Changed 5 years ago by
- Description modified (diff)
comment:2 follow-up: ↓ 3 Changed 5 years ago by
comment:3 in reply to: ↑ 2 Changed 5 years ago by
comment:4 Changed 5 years ago by
- Dependencies set to #21855
- Report Upstream changed from N/A to Fixed upstream, in a later stable release.
I suggest to have one doctest with slightly increased precision for zeta around 1, just to see more than one stieltjes constant.
comment:5 Changed 4 years ago by
- Description modified (diff)
- Priority changed from critical to minor
- Report Upstream changed from Fixed upstream, in a later stable release. to N/A
- Summary changed from Incorrect Series Expansions of zeta(s) around 1 to Document Stieltjes constants
- Type changed from defect to enhancement
See #21963 where I fixed the doctests. I thought it better to use this ticket for the documentation fixes.