ebook9783319927077d4325

Description

Classically developed as a tool for partial differential equations, the¬†analysis of operators known as pseudodifferential analysis is here regarded¬†as a possible help in questions of arithmetic. The operators which make up¬†the main subject of the book can be characterized in terms of congruence¬†arithmetic.¬†They enjoy a Eulerian structure, and are applied to the search¬†for new conditions equivalent to the Riemann hypothesis. These consist in¬†the validity of certain parameter-dependent estimates for a class of Hermitian¬†forms of finite rank. The Littlewood criterion, involving sums of¬†M√∂bius coefficients, and the Weil so-called explicit formula, which leads to¬†his positivity criterion, fit within this scheme, using in the first case Weyl’s¬†pseudodifferential calculus, in the second case Fuchs’.¬† The book should be¬†of interest to people looking for new possible approaches to the Riemann¬†hypothesis, also to newperspectives on pseudodifferential analysis and on¬†the 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.