Riemann hypothesis equivalent

(*start*)
Clear[s]
Limit[(Zeta[s]*Zeta[s]/Zeta[2*s - 1] + Zeta[s]/Zeta[s - 1])/2,
  s -> 1] + Log[2]
Limit[(Zeta[s]*Zeta[s]/Zeta[2*s - 1] + Zeta[s]/Zeta[s - 1])/2,
  s -> 0] + 1/2
% - %%
(*end*)

https://mathoverflow.net/q/400779/25104

2+γ−log4π

(*start*)
Clear[s]
Limit[(Zeta[s]*Zeta[s]/Zeta[2*s - 1] + Zeta[s]/Zeta[s - 1])/2 +
   Zeta[s]*(1 - 1/2^(s - 1)), s -> 0] -
 Limit[(Zeta[s]*Zeta[s]/Zeta[2*s - 1] + Zeta[s]/Zeta[s - 1])/2 +
   Zeta[s]*(1 - 1/2^(s - 1)), s -> 1]
(*end*)

2 + EulerGamma - Log[4 \[Pi]]