Dittrich W. Reassessing Riemann’s Paper ''On the Number of Primes Less Than a Given Magnitude''

New York: Springer, 2018. — xvi, 64 p. — (SpringerBriefs in History of Science and Technology). — ISBN: 978-3-319-91481-7, 978-3-319-91482-4.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.Towards Euler’s Product Formula and Riemann’s Extension of the Zeta Function
Prime Number Counting Function
Riemann as an Expert in Fourier Transforms
On the Way to Riemann’s Entire Function ξ(s)
The Product Representation of ξ(s) and ζ(s)
Derivation of von Mangoldt’s Formula for Ψ(x)
The Number of Roots in the Critical Strip
Riemann’s Zeta Function Regularization
Appendix A: Supplements and Appendix