Math 2.0

Number theory Definition

Riemann zeta function

Builds on Dirichlet seriesHolomorphic functionMeromorphic functionAnalytic continuation

Definition

For a complex number ss and an integer n≥1n \ge 1, let n−s=e−slog⁡nn^{-s} = e^{-s \log n}. For Re⁡s>1\operatorname{Re} s > 1, the Dirichlet series ζ(s)=∑n=1∞1ns\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} converges absolutely, and it defines a holomorphic function on the half-plane Re⁡s>1\operatorname{Re} s > 1. This function has a unique analytic continuation to a meromorphic function on ℂ\C. The Riemann zeta function is this meromorphic function ζ\zeta. Its only pole is a simple pole at s=1s = 1, with residue 11.

Remarks

Example: ζ(2)=π2/6\zeta(2) = \pi^2/6, the value of the series in the Basel problem. Non-example of the series form: at s=1s = 1 the series is the harmonic series, which diverges, and ζ\zeta has its pole there. At s=0s = 0 or s=−1s = -1 the series diverges too, but ζ\zeta is defined there by continuation: ζ(0)=−1/2\zeta(0) = -1/2 and ζ(−1)=−1/12\zeta(-1) = -1/12. For Re⁡s>0\operatorname{Re} s > 0 the continuation is given by ζ(s)=ss−1−s∫1∞x−⌊x⌋xs+1 dx,\zeta(s) = \frac{s}{s - 1} - s \int_1^{\infty} \frac{x - \floor{x}}{x^{s+1}} \, dx, since the integral converges there. The functional equation gives the continuation to all of ℂ\C. The uniqueness of the continuation follows from the identity theorem.

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Analytic continuation, concept, no evidence levelAnalytic continuationconceptDirichlet series, concept, no evidence levelDirichlet seriesconceptHolomorphic function, concept, no evidence levelHolomorphic functionconceptMeromorphic function, concept, no evidence levelMeromorphic functionconceptRiemann zeta function, definition, no evidence levelRiemann zeta functiondefinition
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  1. Analytic continuation concept
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  4. Meromorphic function concept
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Further reading

  1. Web https://en.wikipedia.org/wiki/Riemann_zeta_function
About this page
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MSC11M06
DefinesRiemann zeta function
UsesDirichlet series, Holomorphic function, Meromorphic function, Analytic continuation
Changed2 November 2026

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