Andreas Holmström: New lambda-ring structures in number theory

(joint work with Ane Espeseth and Torstein Vik)

Multiplicative functions are ubiquitous in number theory. Elementary examples include the Euler function, the Liouville function, the Möbius function and many others. More exotic examples arise from zeta functions with Euler products, which may arise from schemes, motives, stacks, Galois representations or modular/automorphic forms. 

We study algebraic structures on the set of all multiplicative functions, and construct in particular four closely related lambda-ring structures on the set of all multiplicative functions. We also suggest a reasonable definition of the "derivative" of a multiplicative function, and we present a new visual and computational tool called a "Tannakian symbol", which can be used for explicit computations of many lambda-ring operations. Taken together, these tools form a unifying framework for a better understanding of many phenomena in number theory and arithmetic geometry.

In the first part of the talk, I will introduce the notion of a lambda-ring and the notion of a multiplicative function, present the basic constructions of our theory, and give some surprising applications to elementary number theory. 

In the second part, I will discuss other applications, including (if time permits) questions about the Grothendieck ring of motives and questions about zeta functions of stacks.