Completed xi function
- Category
- RIEMANN HYPOTHESIS
- Definition
- A symmetric entire version of the zeta function, usually denoted xi or Xi, whose zeros encode the non-trivial zeta zeros.
- Math Level
- TECHNICAL
- Index Excerpt
- xi function; ξ(s); Xi(t); Ξ(t)
The completed xi function is a symmetrised form of the Riemann zeta function, usually written or, on a real spectral variable, .
The completion incorporates additional factors so that the zeta functional equation becomes a particularly clean symmetry. Riemann Console's published sideband work uses products of shifted completed- values because that symmetry is useful in correlation and positivity questions.
See also: Riemann zeta function, functional equation, sideband, positive definiteness..