Opened 2 years ago

Last modified 2 years ago

#21899 new enhancement

Document Stieltjes constants

Reported by: cheuberg Owned by:
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 rws)

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 2 years ago by cheuberg

  • Description modified (diff)

comment:2 follow-up: Changed 2 years ago by rws

See #21963 where I fixed the doctests. I thought it better to use this ticket for the documentation fixes.

comment:3 in reply to: ↑ 2 Changed 2 years ago by behackl

Replying to rws:

See #21963 where I fixed the doctests. I thought it better to use this ticket for the documentation fixes.

Thank you Ralf! We'll polish the documentation of Stieltjes-constants on this ticket.

comment:4 Changed 2 years ago by cheuberg

  • 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 2 years ago by rws

  • 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
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