Reassessing Riemann's paper. On the number of primes less than a given magnitude (Q1649921)

Riemann's article ``Über die Anzahl der Primzahlen unter einer gegebenen Größe'', which appeared in 1859, was the beginning of a long investigation of the connection between the zeros of the Riemann zeta function \(\zeta(s)\) in the critical strip and the distribution of prime numbers. This book (a preliminary version was published online in \url{arXiv:1609.02301}) contains a detailed analysis of Riemann's paper. It begins with a short biography of Riemann, then presents Euler's product representation of \(\zeta(s)\), the extension of the zeta function to the critical strip, the product expansion of Riemann's entire function \(\xi(s)\), Mangoldt's formula, and the number of zeros (called roots in the section title) in the critical strip. The last chapter deals with the technique regularization and connections to physics. The appendix contains calculations that would have benefitted from a few explanations. It also would have been a good idea to include an English translation of Riemann's paper and references to it throughout the text.