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The Theory of Hardy’s Z-Function

Book Description

Hardy's Z-function, related to the Riemann zeta-function (s), was originally utilised by G. H. Hardy to show that (s) has infinitely many zeros of the form +it. It is now amongst the most important functions of analytic number theory, and the Riemann hypothesis, that all complex zeros lie on the line +it, is perhaps one of the best known and most important open problems in mathematics. Today Hardy's function has many applications; among others it is used for extensive calculations regarding the zeros of (s). This comprehensive account covers many aspects of Z(t), including the distribution of its zeros, Gram points, moments and Mellin transforms. It features an extensive bibliography and end-of-chapter notes containing comments, remarks and references. The book also provides many open problems to stimulate readers interested in further research.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. Notation
  7. 1 - Definition of ζ(s), Z(t) and Basic Notions
    1. 1.1 The Basic Notions
    2. 1.2 The Functional Equation for ζ(s)
    3. 1.3 Properties of Hardy’s Function
    4. 1.4 The Distribution of Zeta-Zeros
    5. Notes
  8. 2 - The Zeros on the Critical Line
    1. 2.1 The Infinity of Zeros on the Critical Line
    2. 2.2 A Lower Bound for the Mean Values
    3. 2.3 Lehmer’s Phenomenon
    4. 2.4 Gaps between Consecutive Zeros on the Critical Line
    5. Notes
  9. 3 - The Selberg Class of L-Functions
    1. 3.1 The Axioms of Selberg’s Class
    2. 3.2 The Analogs of Hardy’s and Lindelöf’s Function for S
    3. 3.3 The Degree dF and the Invariants of S
    4. 3.4 The Zeros of Functions in S
    5. Notes
  10. 4 - The Approximate Functional Equations for ζk(s)
    1. 4.1 A Simple AFE for ζ(s)
    2. 4.2 The Riemann-Siegel Formula
    3. 4.3 The AFE for the Powers of ζ(s)
    4. 4.4 The Reflection Principle
    5. 4.5 The AFEs with Smooth Weights
    6. Notes
  11. 5 - The Derivatives of Z(t)
    1. 5.1 The θ and ┌ Functions
    2. 5.2 The Formula for the Derivatives
    3. Notes
  12. 6 - Gram Points
    1. 6.1 Definition and Order of Gram Points
    2. 6.2 Gram’s Law
    3. 6.3 A Mean Value Result
    4. Notes
  13. 7 - The Moments of Hardy’s Function
    1. 7.1 The Asymptotic Formula for the Moments
    2. 7.2 Remarks
    3. Notes
  14. 8 - The Primitive of Hardy’s Function
    1. 8.1 Introduction
    2. 8.2 The Laplace Transform of Hardy’s Function
    3. 8.3 Proof of Theorem 8.2
    4. 8.4 Proof of Theorem 8.3
    5. Notes
  15. 9 - The Mellin Transforms of Powers of Z(t)
    1. 9.1 Introduction
    2. 9.2 Some Properties of the Modified Mellin Transforms
    3. 9.3 Analytic Continuation of Mk(s)
    4. Notes
  16. 10 - Further Results on Mk(s) and Zk(s)
    1. 10.1 Some Relations for Mk(s)
    2. 10.2 Mean Square Identities for Mk(s)
    3. 10.3 Estimates for Mk(s)
    4. 10.4 Natural Boundaries
    5. Notes
  17. 11 - On Some Problems Involving Hardy’s Function and Zeta-moments
    1. 11.1 The Distribution of Values of Hardy’s Function
    2. 11.2 The Order of the Primitive of Hardy’s Function
    3. 11.3 The Cubic Moment of Hardy’s Function
    4. 11.4 Further Problems on the Distribution of Values
    5. Notes
  18. References
  19. Author Index
  20. Subject Index