Zeta functions associated to polyhedral cones

• Grant number: 25/06117-4
• Support Opportunities: Scholarships in Brazil - Master
• Start date: November 01, 2025
• End date: February 28, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Sinai Robins
• Grantee: André Rosenbaum Coelho
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/03167-5 - Classical, asymptotic, quantum and geometric combinatorics, AP.TEM

Abstract

We study certain zeta functions associated to polyhedral cones, and in particular Shintani zetafunctions. One of our main tools is a new summation formula, namely the cone Lipschitz summation formula, which extends the classical Lipschitz summation formula on the integers, to a summation formula on the d-dimensional integer lattice, by summing over the integer points in a cone. The cone Lipschitz summation formula may also be viewed as an extension of the Hurwitz zeta function, to several variables. It is alsorelated to the Shintani zeta function. We use the cone Lipschitz summation formula to derive Fourierexpansions, as well as some related arithmetic properties of cones, and of divisor functions.