original_query: What is the current knowledge about the irrationality of ζ(5)?
query: What is the current knowledge about the irrationality of ζ(5)?
answer: It is unknown whether ζ(5) is irrational.  While it's been shown that infinitely many of the numbers ζ(2n+1) are irrational, and at least one of ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational,  a proof for the irrationality of ζ(5) specifically remains elusive.  Attempts to extend Apéry's method have been unsuccessful due to the size of the coefficients involved.