Introduction

Abstract

Most practitioners of pseudo-differential analysis and of analytic number theory would probably regard the two fields as being as far apart as conceivable. However, we wish here to convey the idea that, if deepened in its appropriate aspects, pseudo-differential analysis (mostly, but not only, one-dimensional pseudodi fferential analysis in this book) may find a place in the bag of tools of modular form theory. To our PDE colleagues, we shall simply offer the apology that doing some export cannot hurt: more seriously, we have written this book under the assumption that some readers with very little, or no previous knowledge of automorphic function theory, might wish to find a reason to approach this fascinating domain, in which exact formulas of much aesthetic appeal are often the reward of spectral-theoretic questions. Analysts may also find, in the first chapter, aspects of pseudo-differential analysis unknown to them.