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Zeta-Regularized Sum

The zeta-regularized sum of a positive sequence {lambda_n} is obtained from the analytic continuation of its spectral zeta function

zeta_lambda(s)=sum_(n=1)^inftylambda_n^(-s).

If zeta_lambda(s) has an analytic continuation that is regular at s=-1, the zeta-regularized value of sum_(n)lambda_n is defined to be zeta_lambda(-1).

For lambda_n=n, this prescription gives

1+2+3+...=^(zeta)zeta(-1)=-1/(12).

This equality is a statement about regularization, not convergence of the ordinary partial sums. Zeta-regularized sums are related to, but distinct from zeta-regularized products, which use -zeta_lambda^'(0).

See also

Analytic Continuation, Regularized Sum, Riemann Zeta Function, Zeta-Regularized Product

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References

Elizalde, E. Ten Physical Applications of Spectral Zeta Functions. Berlin, Germany: Springer-Verlag, 1995.

Cite this as:

Weisstein, Eric W. "Zeta-Regularized Sum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Zeta-RegularizedSum.html

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