Reassessing Riemann's Paper: On the Number of Primes Less Than a Given Magnitude (SpringerBriefs in History of Science and Technology)

In this book, the author pays tribute to Bernhard Riemann (1826-1866), a mathematician with revolutionary ideas, whose work on the theory of integration, the Fourier transform, the hypergeometric differential equation, etc. contributed immensely to mathematical physics. The text concentrates in particular on Riemann’s only work on prime numbers, including ideas – new at the time – such as analytical continuation into the complex plane and the product formula for entire functions. A detailed analysis of the zeros of the Riemann zeta-function is presented. The impact of Riemann’s ideas on regularizing infinite values in field theory is also emphasized. This revised and enhanced new edition contains three new chapters, two on the application of Riemann’s zeta-function regularization to obtain the partition function of a Bose (Fermi) oscillator and one on the zeta-function regularization in quantum electrodynamics. Appendix A2 has been re-written to make the calculations more transparent. A summary of Euler-Riemann formulae completes the book. Preface to the Second Edition Preface to the First Edition Contents 1 Towards Euler's Product Formula and Riemann's Extension of the Zeta Function 2 Prime Number Counting Function 3 Riemann as an Expert in Fourier Transforms 4 On the Way to Riemann's Entire Function ξ(s) 5 The Product Representation of ξ(s) and ζ(s) by Riemann (1859) and Hadamard (1893) 6 Derivation of von Mangoldt's Formula for Ψ(x) 7 The Number of Roots in the Critical Strip 8 Riemann's Zeta Function Regularization 9 ζ-Function Regularization of the Partition Function of the Harmonic Oscillator 10 ζ-Function Regularization of the Partition Function of the Fermi Oscillator 11 The Zeta Function in Quantum Electrodynamics (QED) Summary of Euler-Riemann Formulae Appendix A Supplements and Appendix A.1 Supplements A.1.1 The Origins of the Functional Equation for Dirichlet's Function A.1.2 Riemann's Functional Equation A.1.3 What Is a Function? A.2 Appendix A.2.1 Remarks on the Oscillatory Behavior of the Prime Counting Function Acknowledgements References