Reassessing Riemann's Paper

1. Front Matter

   Pages i-xix

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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. Back Matter

    Pages 45-65

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In this book, the author pays tribute to Bernhard Riemann (1826–1866), mathematician with revolutionary ideas, whose work on the theory of integration, the Fourier transform, the hypergeometric differential equation, etc. contributed immensely to mathematical physics. This book concentrates in particular on Riemann’s only work on prime numbers, including such then new ideas as analytical continuation in 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.

• Riemann's Paper on Prime Numbers
• Zeta-function regularization
• Functional equation
• Fourier transform
• Riemann's function
• Mangoldt's formula
• Zeros of zeta function and critical strip
• History of prime number theory

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

  Walter Dittrich

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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