Riemann zeta function and the best constants of three series of Sobolev inequalities(Theory,Applied Integrable Systems,<Special Issue>Joint Symposium of JSIAM Activity Groups 2007)

Abstract

We clarified the variational meaning of the special values ζ(2M)(M=1,2,3,…) of Riemann zeta function ζ(s). These are essentially the best constants of three series of Sobolev inequalities. In the background we have three kind of boundary value problem (periodic, Dirichlet, Neumann) for the simplest differential operator (-1)^M(d/dx)^<2M>. Green functions for these boundary value problems can be given by Bernoulli polynomials. Sobolev energy of G(x,y) with respect to x is its diagonal value G(y,y). Its maximum max__<|y|≤1> G(y,y) is equal to the best constant for corresponding Sobolev inequality.