Pseudodifferential Methods in Number Theory

Classically developed as a tool for partial differential equations, theanalysis of operators known as pseudodifferential analysis is here regardedas a possible help in questions of arithmetic. The operators which make upthe main subject of the book can be characterized in terms of congruencearithmetic.They enjoy a Eulerian structure, and are applied to the searchfor new conditions equivalent to the Riemann hypothesis. These consist inthe validity of certain parameter-dependent estimates for a class of Hermitianforms of finite rank. The Littlewood criterion, involving sums ofMöbius coefficients, and the Weil so-called explicit formula, which leads tohis positivity criterion, fit within this scheme, using in the first case Weyl'spseudodifferential calculus, in the second case Fuchs'. The book should beof interest to people looking for new possible approaches to the Riemannhypothesis, also to newperspectives on pseudodifferential analysis and onthe way it combines with modular form theory. Analysts will have no difficulty with the arithmetic aspects, with which, save for very few exceptions,no previous acquaintance is necessary.