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The Riemann zeta function on the half-plane
Definition
For with , define the Riemann zeta function by
Here is the complex exponential and is the real logarithm of the positive integer . The preceding lemma The Dirichlet series for zeta converges absolutely and locally uniformly on the half-plane ↗ proves that this Dirichlet series converges absolutely and locally uniformly on the open half-plane , so the definition is well posed exactly on that domain.
Depends on
Used by
- The defining Dirichlet series for zeta diverges at s=1 because it becomes the harmonic series Counterexample
- The completed zeta function Λ(s)=π^-s/2Γ(s/2)ζ(s) Definition
- The level-one Eisenstein series Eₖ and the weight-two series E₂ Definition
- A short Euler-product truncation already numerically approximates zeta at 2 Example
- The special-value formula gives ζ(4)=π⁴/90 Example
- FALSE: the classical functional equation alone characterizes the Riemann zeta function False statement
- The Dirichlet series for zeta converges absolutely and locally uniformly on the half-plane Res>1 Lemma
- Eisenstein series are modular forms; their Fourier coefficients Theorem
- For Res>0, zeta admits the fractional-part integral formula with a simple residue-one pole at 1 Theorem
- The completed zeta function has its Mellin-theta integral representation on Res>1 Theorem
- The Dirichlet eta series is holomorphic on Res>0 and equals the prefactor times zeta there Theorem
- The only zeros of zeta on the nonpositive real axis are the negative even integers, and every other zero lies in the open critical strip Theorem
- The principal Dirichlet L-function factors through zeta Theorem
- The Riemann xi function is entire of order one, real on the real axis, and symmetric under s↦1-s Theorem
- The Riemann zeta function has its Euler product on the half-plane Res>1 Theorem
Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 6 §2 (standard reference, not scraped)
- K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 11 §3 (standard reference, not scraped)