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 Möbius 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.
