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 the Epstein's zeta function, at least at any real s>n/2. We deduce from this a new proof of Sarnak and Str\"ombergsson's theorem asserting that the root lattices D4 and E8, as well as the Leech lattice, achieve a strict local minimum of the 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).