Multiplicative Analysis

A prime number is a positive integer that is divisible only by 1 and itself: the first five prime numbers are 2, 3, 5, 7, and 11. The prime numbers are the building blocks needed to construct the other positive integers using multiplication. In the same way that a water molecule consists of two hydrogen atoms and one oxygen atom, the number 12 is comprised two 2s and one 3, since 12 = 2 x 2 x 3. Euclid demonstrated already in antiquity that there is an infinite number of prime numbers, but our understanding of their distribution among the integers remains rudimentary. This question concerns the interplay between the additive and multiplicative structures of the integers. In this project, we will study the prime numbers from two different perspectives. In the first perspective, we associate a signal with a given frequency to each prime number, where the frequency depends on the size of the prime number. In contrast to the classical periodic functions that we obtain from physical systems, the functions we obtain by combining the prime number frequencies are almost periodic. In the second perspective, we associate each prime number to an independent variable, which leads to functions of an infinite number of variables. It turns out that the two perspectives are in many ways equivalent, and it is through this equivalence we hope to obtain a better understanding of the prime numbers and their properties.

Multiplicative analysis is an emergent field of analysis that lies in the intersection of complex analysis, operator theory, and analytic number theory. The main objective of the research project is to solve problems reinforcing the interplay between these different branches of analysis. Of particular interest are interactions between multiplicative analysis and the Riemann zeta function. The central objects in the field are the Hardy spaces of Dirichlet series, and our main objective is to study analogues of classical results and structures for these objects. In the Hilbertian setting, we aim at breaking out of the current crop of local results to improve our understanding of global Carleson measures and the global structure of zero sets. Our approach to these problems is based on the theory of almost periodic functions and certain properties of the Riemann zeta function. In the non-Hilbertian setting, our selection of problems revolves around the local embedding problem. The focus is on dual spaces, interpolating sequences, and embeddings (both contractive and non-contractive) between function spaces. We will use interdisciplinary techniques ranging from infinite-dimensional harmonic analysis, via function theory of one and several variables, to analytic number theory and multiplicative chaos. The 4-year project will support 6,33 man-years of research carried out by the project manager and one PhD student. A significant part of the project is the long-term hybrid "Multiplicative analysis seminar" that will attract researchers from analytic number theory and operator-related function theory to interdisciplinary research in multiplicative analysis.