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The Claim

By analytic continuation, the Riemann zeta function satisfies ζ(-1) = -1/12.

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The Short Version

Standard mathematical references agree that the analytically continued Riemann zeta function has value ζ(-1) = -1/12. This does not mean the ordinary series 1+2+3+4+⋯ converges to -1/12; it means the unique analytic extension of ζ(s) to s = -1 takes that value.

Caveats

  • This value belongs to the analytically continued zeta function, not to the ordinary sum of 1+2+3+4+⋯ in the usual convergence sense.
  • The original Dirichlet series definition of ζ(s) converges only for Re(s) > 1, so evaluating at s = -1 requires analytic continuation.
  • Popular presentations often blur the distinction between regularized values and ordinary sums; that distinction is essential here.

The Receipts

  1. Riemann zeta function

    Encyclopaedia Britannica

  2. Bernoulli Number

    MathWorld

  3. Analytic Continuation of the Riemann Zeta Function

    University of Oklahoma (nhn.ou.edu)

  4. Particular values of the Riemann zeta function

    Wikipedia

  5. Riemann zeta function

    Wikipedia

  6. 1 + 2 + 3 + 4 + ⋯

    Wikipedia

  7. The Euler-Maclaurin formula, Bernoulli numbers, the zeta function and real variable analytic continuation

    terrytao.wordpress.com

  8. Analytic continuation of Riemann's zeta function and values at negative integers

    Proceedings of the American Mathematical Society (ams.org)

  9. Riemann ζ function

    OEIS Wiki

  10. Riemann Zeta Function

    Wolfram MathWorld

+ 16 more sources — see the full list on Lenz

Filed Under

Riemann Zeta Function

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