The prime number theorem, which characterizes the asymptotic distribution of prime numbers, has a long history in terms of its different proofs and the ideas about them. The first proof was given by Hadamard and de la Vallée Poussin in 1896, which was not elementary and made use of complex analysis applied to the Riemann zeta function ς(s). Several simplified proofs were given later especially by Landau and Wiener. In 1921 Hardy was the first to ask whether it is possible to reasonable to expect an elementary proof, one that is not fundamentally dependent on the theory of complex functions. In the year 1948 the mathematical world was stunned to witness a truly elementary proof which used only the simplest properties of the logarithm function, by Erdős and Selberg, which also led to a dispute between them later.
In this project we are going to investigate the basic ideas about this innovative and truly original elementary proof, in particular the famous Selberg’s inequality and Erdős’ ingenious derivation of the crucial asymptotic identity pn~pn+1 directly from it. Along the way of reading some original papers, we will also learn and review some basic knowledge in number theory and enumerative combinatorics, as well as in calculus and asymptotic analysis.
Obs! För GU-studenter räknas projektet som ett projekt i Matematik (MMG900/MMG910).
Förkunskapskrav Calculus or basic course in analysis that everybody has; interest and curiosity about prime numbers are appreciated; what’s more, a bit of maturity or confidence about how to count a set in a clever way is even better. Basic knowledge in elementary number theory helps but is not necessary.
Examinator Maria Roginskaya, Marina Axelson-Fisk