cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A380908 Decimal expansion of lim_{s->1} (zeta(s) - Pi^(s/2)/((s-1)*Gamma(s/2))) (negated).

Original entry on oeis.org

9, 7, 6, 9, 0, 4, 2, 9, 1, 0, 3, 3, 8, 7, 8, 9, 6, 6, 1, 8, 5, 6, 8, 9, 7, 5, 2, 0, 9, 3, 5, 0, 4, 7, 0, 8, 3, 7, 8, 0, 6, 7, 8, 7, 2, 8, 4, 7, 9, 4, 9, 2, 4, 0, 4, 7, 4, 6, 0, 7, 9, 2, 7, 7, 8, 7, 0, 2, 8, 6, 4, 3, 5, 2, 3, 2, 7, 5, 4, 2, 0, 0, 2, 9, 2, 0, 1, 4, 3, 0, 4, 8, 8, 2, 9
Offset: 0

Views

Author

Peter Luschny, Mar 04 2025

Keywords

Comments

This limit is Mathlib's definition of the 'completed Riemann zeta function' at s = 1. Mathematically zeta(1) is undefined; for the 'completed zeta function' the above limit value is assigned by construction (see Loeffler&Stoll). Such a value is also called a 'junk value' in several proof systems.

Examples

			-0.976904291033878966185689752093504708378...
		

Crossrefs

Programs

  • Maple
    c := -(gamma - log(4*Pi))/2: evalf(c, 110)*10^95: ListTools:-Reverse(convert(floor(%), base, 10));
  • Mathematica
    First[RealDigits[(EulerGamma - Log[4*Pi])/2, 10, 100]] (* Paolo Xausa, Mar 05 2025 *)
  • PARI
    (Euler-log(4*Pi))/2 \\ Charles R Greathouse IV, Sep 03 2025

Formula

Equals (gamma - log(4*Pi))/2.
Equals gamma + Psi(1/2)/2 - log(Pi^(1/2)) = A001620 - A114864 - A155968.
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