Despite this, many mathematicians working in this area expect a breakthrough sometime soon. Indeed, recent work by Wadim Zudilin and Tanguy Rivoal has shown that infinitely many of the numbers ζ(2n + 1) must be irrational, and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational. Their work uses linear forms in values of the zeta function and estimates upon them to bound the dimension of a vector space spanned by values of the zeta function at odd integers. Hopes that Zudilin could cut his list further to just one number did not materialise, but work on this problem is still an active area of research. Higher zeta constants have application to physics: they describe correlation functions in quantum spin chains. References External links Zeta and L-functions Theorems in number theory