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Riemann's Zeta Function

Harold M. Edwards
Dover Publications
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a historically-driven text on the Riemann zeta function. Its starting point is Riemann’s original 1859 paper in which he introduced the zeta function, indicated how it could be used to prove the prime number theorem, and asserted several of its properties. The book is well-written and easy to follow, and has many interesting historical digressions. The present volume is a Dover 2001 unaltered reprint of the 1974 edition from Academic Press.

The book is not comprehensive but deals with six “questions unresolved by Riemann” (pp. 37–38). The questions have to do with the location and density of the zeroes of the zeta function, some functional properties, and finally the prime number theorem and the Riemann hypothesis.

The book works forward through the history of subsequent developments on these six questions. The treatment is eclectic, but is biased towards the original proofs rather than newer, slicker proofs. There are three chapters on numerical methods that are used in calculating zeroes of the zeta function. Edwards is one of those who urges us to read the masters, not the pupils; in this vein, the book includes an English translation of Riemann’s paper in an appendix.

Bottom line: not comprehensive or (at this point) up-to-date, but you will learn a lot about the zeta function if you work through the book. A good monograph that goes into much greater depth is Ivić’s The Riemann Zeta Function: Theory and Applications.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.


Chapter 1. Riemann's Paper
  1.1 The Historical Context of the Paper
  1.2 The Euler Product Formula
  1.3 The Factorial Function
  1.4 The Function zeta (s)
  1.5 Values of zeta (s)
  1.6 First Proof of the Functional Equation
  1.7 Second Proof of the Functional Equation
  1.8 The Function xi (s)
  1.9 The Roots rho of xi
  1.10 The Product Representation of xi (s)
  1.11 The Connection between zeta (s) and Primes
  1.12 Fourier Inversion
  1.13 Method for Deriving the Formula for J(x)
  1.14 The Principal Term of J(x)
  1.15 The Term Involving the Roots rho
  1.16 The Remaining Terms
  1.17 The Formula for pi (x)
  1.18 The Density dJ
  1.19 Questions Unresolved by Riemann
Chapter 2. The Product Formula for xi
  2.1 Introduction
  2.2 Jensen's Theorem
  2.3 A Simple Estimate of absolute value of |xi (s)|
  2.4 The Resulting Estimate of the Roots rho
  2.5 Convergence of the Product
  2.6 Rate of Growth of the Quotient
  2.7 Rate of Growth of Even Entire Functions
  2.8 The Product Formula for xi
Chapter 3. Riemann's Main Formula
  3.1 Introduction
  3.2 Derivation of von Mangoldt's formula for psi (x)
  3.3 The Basic Integral Formula
  3.4 The Density of the Roots
  3.5 Proof of von Mangoldt's Formula for psi (x)
  3.6 Riemann's Main Formula
  3.7 Von Mangoldt's Proof of Reimann's Main Formula
  3.8 Numerical Evaluation of the Constant
Chapter 4. The Prime Number Theorem
  4.1 Introduction
  4.2 Hadamard's Proof That Re rho<1 for All rho
  4.3 Proof That psi (x) ~ x
  4.4 Proof of the Prime Number Theorem
Chapter 5. De la Vallée Poussin's Theorem
  5.1 Introduction
  5.2 An Improvement of Re rho<1
  5.3 De la Vallée Poussin's Estimate of the Error
  5.4 Other Formulas for pi (x)
  5.5 Error Estimates and the Riemann Hypothesis
  5.6 A Postscript to de la Vallée Poussin's Proof
Chapter 6. Numerical Analysis of the Roots by Euler-Maclaurin Summation
  6.1 Introduction
  6.2 Euler-Maclaurin Summation
  6.3 Evaluation of PI by Euler-Maclaurin Summation. Stirling's Series
  6.4 Evaluation of zeta by Euler-Maclaurin Summation
  6.5 Techniques for Locating Roots on the Line
  6.6 Techniques for Computing the Number of Roots in a Given Range
  6.7 Backlund's Estimate of N(T)
  6.8 Alternative Evaluation of zeta'(0)/zeta(0)
Chapter 7. The Riemann-Siegel Formula
  7.1 Introduction
  7.2 Basic Derivation of the Formula
  7.3 Estimation of the Integral away from the Saddle Point
  7.4 First Approximation to the Main Integral
  7.5 Higher Order Approximations
  7.6 Sample Computations
  7.7 Error Estimates
  7.8 Speculations on the Genesis of the Riemann Hypothesis
  7.9 The Riemann-Siegel Integral Formula
Chapter 8. Large-Scale Computations
  8.1 Introduction
  8.2 Turing's Method
  8.3 Lehmer's Phenomenon
  8.4 Computations of Rosser, Yohe, and Schoenfeld
Chapter 9. The Growth of Zeta as t --> infinity and the Location of Its Zeros
  9.1 Introduction
  9.2 Lindelöf's Estimates and His Hypothesis
  9.3 The Three Circles Theorem
  9.4 Backlund's Reformulation of the Lindelöf Hypothesis
  9.5 The Average Value of S(t) Is Zero
  9.6 The Bohr-Landau Theorem
  9.7 The Average of absolute value |zeta(s)| superscript 2
  9.8 Further Results. Landau's Notation o, O
Chapter 10. Fourier Analysis
  10.1 Invariant Operators on R superscript + and Their Transforms
  10.2 Adjoints and Their Transforms
  10.3 A Self-Adjoint Operator with Transform xi (s)
  10.4 The Functional Equation
  10.5 2 xi (s)/s(s - 1) as a Transform
  10.6 Fourier Inversion
  10.7 Parseval's Equation
  10.8 The Values of zeta (-n)
  10.9 Möbius Inversion
  10.10 Ramanujan's Formula
Chapter 11. Zeros on the Line
  11.1 Hardy's Theorem
  11.2 There Are at Least KT Zeros on the Line
  11.3 There Are at Least KT log T Zeros on the Line
  11.4 Proof of a Lemma
Chapter 12. Miscellany
  12.1 The Riemann Hypothesis and the Growth of M(x)
  12.2 The Riemann Hypothesis and Farey Series
  12.3 Denjoy's Probabilistic Interpretation of the Riemann Hypothesis
  12.4 An Interesting False Conjecture
  12.5 Transforms with Zeros on the Line
  12.6 Alternative Proof of the Integral Formula
  12.7 Tauberian Theorems
  12.8 Chebyshev's Identity
  12.9 Selberg's Inequality
  12.10 Elementary Proof of the Prime Number Theorem
  12.11 Other Zeta Functions. Weil's Theorem
Appendix. On the Number of Primes Less Than a Given Magnitude (By Bernhard Riemann)
  References; Index