# analytic number theory

Analytic number theory^{} 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 theorem^{}.

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 (http://planetmath.org/EulerMaclaurinSummationFormula), Abelâ€™s lemma (summation by parts^{}), 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 Dirichletâ€™s contributions to analytic number theory is the Dirichlet series. As an example, the Dirichlet series of a Dirichlet character^{} is a Dirichlet L-series. A tool that is helpful for studying any Dirichlet series is the Euler product^{}. The most famous Dirichlet series is the Riemann zeta function^{}, which is the Dirichlet series of the completely multiplicative function^{} $1$. 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 $\frac{1}{2}$. Its to prime numbers^{} is made clearer by the Euler product formula.

Title | analytic number theory |
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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 |