Spherical designs and zeta functions of lattices

We set up a connection between the theory of spherical designs and the question of minima of Epstein's zeta function. More precisely, we prove that a Euclidean lattice, all layers of which hold a 4-design, achieves a local minimum of Epstein's zeta function, at least at any real s > n/2. We deduce from this a new proof of Sarnak and Strombergsson's theorem asserting that the root lattices D-4 and E-8, as well as the Leech lattice Lambda(24), achieve a strict local minimum of Epstein's zeta function at any s > 0. Furthermore, our criterion enables us to extend their theorem to all the so-called extremal modular lattices ( up to certain restrictions) using a theorem of Bachoc and Venkov, and to other classical families of lattices ( e. g., the Barnes-Wall lattices).