Riemann zeta function and the best constants of five series ofSobolev inequalities

We clarified the variational meaning of the special values ζ(2M) (M = 1, 2, 3, · · · ) of Riemann zeta function ζ(z). These are essentially the best constants of five series of Sobolev inequalities. In the background, we consider five kinds of boundary value problem (periodic, antiperiodic, Dirichlet, Neumann, Dirichlet-Neumann) for a differential operator (−1) (d/dx) . Green functions for these boundary value problems are given by Bernoulli polynomials. Green functions are simultaneously reproducing kernels for certain Hilbert spaces. Applying Schwarz inequality to the reproducing relation, we have found the best constants of Sobolev inequalities. § 1. Conclusion We consider the following five cases: P (Periodic), AP (Anti Periodic), D (Dirichlet), N (Neumann), DN (Dirichlet-Neumann). Received December 29, 2008. Accepted February 6, 2009. 2000 Mathematics Subject Classification(s): Primary 34B27, Secondary 46E35.