An Epic Drama: The Development of the Prime Number Theorem - Benjamin Fine; Gerhard Rosenberger (Scientia Series A: Mathematical Sciences, 2025)
The prime number theorem, describing the aymptotic density of the prime numbers, has often been touted as the most surprising result in mathematics. The statement and development of the theorem by Legendre, Gauss and others and its eventual proof by Hadamard and de al Vall´ee-Poussin span the whole nineteenth century and encompass the growth of a brand new field in analytic number theory. As an outgrowth of the techniques of the proof is the Riemann hypothesis which today is perhaps the outstanding open problem in mathematics. These ideas and occurences certainly constitute an epic drama within the history of mathematics and one that is not as well known among the general mathematical community as it should be. In the present paper we trace out the paper, the development of the proof and a raft of other ideas, results and concepts that come from the prime number theorem.
Keywords: Prime number theorem; zeta functions; Chebyshev’s Theorem; Riemann Hypothesis; Dirichlet Series.