Opened 4 years ago

Last modified 4 years ago

## #21899 new enhancement

# Incorrect Series Expansions of zeta(s) around 1 — at Initial Version

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: | Stopgaps: |

### Description

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.

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)."

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