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Reassessing Riemann's Paper

On the Number of Primes Less Than a Given Magnitude

  • Book
  • © 2021

Overview

Authors:
  1. Walter Dittrich

    1. Institut für Theoretische Physik, Universität Tübingen, Tübingen, Germany

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  • Didactically improved presentation of selected applications of Riemann’s ?-function
  • Provides three new chapters and a summary of Euler-Riemann formulae
  • Emphasizes the importance of Riemann’s functional equation for mathematics and physics

Part of the book series: SpringerBriefs in History of Science and Technology (BRIEFSHIST)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xi
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  2. Towards Euler’s Product Formula and Riemann’s Extension of the Zeta Function

    • Walter Dittrich
    Pages 1-6

  3. Prime Number Counting Function

    • Walter Dittrich
    Pages 7-12

  4. Riemann as an Expert in Fourier Transforms

    • Walter Dittrich
    Pages 13-16

  5. On the Way to Riemann’s Entire Function \(\xi (s)\)

    • Walter Dittrich
    Pages 17-23

  6. The Product Representation of \(\xi (s)\) and \(\zeta (s)\) by Riemann (1859) and Hadamard (1893)

    • Walter Dittrich
    Pages 25-27

  7. Derivation of von Mangoldt’s Formula for \(\Psi (x)\)

    • Walter Dittrich
    Pages 29-32

  8. The Number of Roots in the Critical Strip

    • Walter Dittrich
    Pages 33-37

  9. Riemann’s Zeta Function Regularization

    • Walter Dittrich
    Pages 39-43

  10. ζ-Function Regularization of the Partition Function of the Harmonic Oscillator

    • Walter Dittrich
    Pages 45-50

  11. ζ-Function Regularization of the Partition Function of the Fermi Oscillator

    • Walter Dittrich
    Pages 51-56

  12. The Zeta Function in Quantum Electrodynamics (QED)

    • Walter Dittrich
    Pages 57-68

  13. Back Matter

    Pages 69-107
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Keywords

About this book

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.



Authors and Affiliations

  • Institut für Theoretische Physik, Universität Tübingen, Tübingen, Germany

    Walter Dittrich

About the author

Prof. Dr. Walter Dittrich was head of the quantum electrodynamics group at the University of Tübingen before his retirement in 2001 and is still actively publishing papers and books in classical and quantum physics. He received his doctorate under Prof. Heinz Mitter at Heisenberg’s institute in Munich and continued pre- and postdoc work at Brown University, Harvard and MIT. He profited immensely from lectures by and discussions with Profs. Herb Fried, Ken Johnson, Steve Weinberg, Julian Schwinger and, later on, at the Institute for Advanced Study (IAS) in Princeton, Steve Adler and David Gross at Princeton University. He started his work on gauge theories and QED in collaboration with Schwinger in the late 1960s. He was visiting professor at UCLA, Berkeley, Stanford and the IAS. He has over 30 years of teaching experience and is one of the key scientists in developing the theoretical framework of quantum electrodynamics.

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