analytic number theory

Analytic number theoryMathworldPlanetmath uses the machinery of analysis to tackle questions related to integers and transcendence. One of its most famous achievements is the proof of the prime number theoremMathworldPlanetmath.

One concept that is important in analytic number theory is asymptotic estimates. Tools that are used to obtain asymptotic estimates for sums include the Euler-Maclaurin summation formula (, Abel’s lemma (summation by partsPlanetmathPlanetmath), the convolution method, and the Dirichlet hyperbola method. Asymptotic estimates are important for determining asymptotic densities of certain subsets of the natural numbers.

Another one of Dirichlet’s contributions to analytic number theory is the Dirichlet series. As an example, the Dirichlet series of a Dirichlet characterDlmfMathworldPlanetmath is a Dirichlet L-series. A tool that is helpful for studying any Dirichlet series is the Euler productMathworldPlanetmath. The most famous Dirichlet series is the Riemann zeta functionDlmfDlmfMathworldPlanetmath, which is the Dirichlet series of the completely multiplicative functionMathworldPlanetmath 1. This leads up to what is possibly the most important unsolved problem in analytic number theory: the Riemann hypothesis. This that all nontrivial zeros of the Riemann zeta function have real part equal to 12. Its to prime numbersMathworldPlanetmath is made clearer by the Euler product formula.

Title analytic number theory
Canonical name AnalyticNumberTheory
Date of creation 2013-03-22 15:59:55
Last modified on 2013-03-22 15:59:55
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 8
Author Wkbj79 (1863)
Entry type Topic
Classification msc 11N37
Classification msc 11M06
Classification msc 11N05
Classification msc 11-01