ξ functional equation

ξ functional equation

Every non-trivial zero of the Riemann zeta function ζ(s) has real part 1/2. Equivalent forms involve the ξ-function, the explicit formula, and zero-density estimates on the critical strip.

ξ functional equation —lemma→ Riemann

ξ(s) := ½ s(s−1) π^{−s/2} Γ(s/2) ζ(s) is entire of order 1, obeys ξ(s)=ξ(1−s), and its zeros are exactly the non-trivial —lemma→ ξ functional equation

ξ functional equation —implies→ Critical line