Pseudodifferential Methods in Number Theory

  • André Unterberger

Part of the Pseudo-Differential Operators book series (PDO, volume 13)

Table of contents

  1. Front Matter
    Pages i-vi
  2. André Unterberger
    Pages 1-6
  3. André Unterberger
    Pages 7-15
  4. André Unterberger
    Pages 17-47
  5. André Unterberger
    Pages 91-104
  6. André Unterberger
    Pages 105-125
  7. André Unterberger
    Pages 127-140
  8. André Unterberger
    Pages 141-166
  9. Back Matter
    Pages 167-173

About this book


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 coeffcients, 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 new perspectives on pseudodifferential analysis and on the way it combines with modular form theory. Analysts will have no diffculty with the arithmetic aspects, with which, save for very few exceptions, no previous acquaintance is necessary.


pseudodifferential analysis in arithmetic approach to the zeros of Riemann's zeta function modular distribution theory Weyl and Fuchs pseudodifferential calculi number theory

Authors and affiliations

  • André Unterberger
    • 1
  1. 1.Department of MathematicsUniversity of ReimsReimsFrance

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