Browse Teach Curate

Signs of Higher-Order Derivatives of Epstein Zeta and Theta Functions

Extends classical first-derivative sign results (Rankin, Cassels, Ennola, Diananda, Montgomery) to determine the signs of all higher-order derivatives of the Epstein zeta and lattice theta functions parametrized over the upper half-plane. The monotonicity and convexity these signs encode are then used to attack lattice-energy minimization problems, where they help identify the optimal lattice.

Marvin Text

SIGNS OF HIGH ORDER DERIVATIVES FOR THE THETA AND EPSTEIN ZETA FUNCTIONS AND APPLICATION KAIXIN

For a planar lattice of unit cell area, the Epstein zeta function (a sum of inverse even powers of nonzero lattice-vector lengths) and the theta function (a Gaussian sum over lattice vectors) are the…