Hi, here, in wikipedia, the Landau-Ramanujan constant appears under a list of suspected transcendentals. I could not find anywhere a statement or a proof of it's irrationality. So, my question is, is the constant irrational( since it's under a list of suspected transcendentals, is it right to assume that it is irrational?) And if it is, is there a proof available? Thanks in advance.
This is not a definitive answer, just a few links. This link of "important irrational constants" lists it as irrational, without a citation. The Encyclopedia of Integer Sequences has quite a few references at the page for Landau-Ramanujan, one or more of which may contain what you seek.