Book Review: Analytic number theory

Analytic Number Theory

Analytic Number Theory

What is number theory? One might have thought that it was simply the study of numbers, but that is too broad a definition, since numbers are almost ubiquitous in mathematics. To see what distinguishes number theory from the rest of mathematics, let us look at the equation x + y = 15925, and consider whether it has any solutions. One answer is that it certainly does: indeed, the solution set for...

Basic Analytic Number Theory

We give an informal introduction to the most basic techniques used to evaluate moments on the critical line of the Riemann zeta-function and to find asymptotics for sums of arithmetic functions.

Topics in Analytic Number Theory

An arithmetic function is a function f : N → C; there are many interesting and natural examples in analytic number theory. To begin with we consider what is perhaps the best known π(n) := xn: 1 P (x), the usual counting function of the primes. Various heuristic arguments suggest that one should expect x to be prime with probability 1/ log x, and coupled with a body of numerical evidence this pr...

Poincaré and Analytic Number Theory

Un domaine arithmétique où l’unité semble faire absolument défaut, c’est la théorie des nombres premiers ; on n’a trouvé que des lois asymptotiques et l’on n’en doit pas espérer d’autres; mais ces lois sont isolées et l’on n’y peut parvenir que par des chemins différents qui ne semblent pas pouvoir communiquer entre eux. Je crois entrevoir d’où sortira l’unité souhaitée, mais je ne l’entrevois ...