## Multiplicative Number Theory I: Classical TheoryPrime numbers are the multiplicative building blocks of natural numbers. Understanding their overall influence and especially their distribution gives rise to central questions in mathematics and physics. In particular their finer distribution is closely connected with the Riemann hypothesis, the most important unsolved problem in the mathematical world. Assuming only subjects covered in a standard degree in mathematics, the authors comprehensively cover all the topics met in first courses on multiplicative number theory and the distribution of prime numbers. They bring their extensive and distinguished research expertise to bear in preparing the student for intelligent reading of the more advanced research literature. This 2006 text, which is based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State, is enriched by comprehensive historical notes and references as well as over 500 exercises. |

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### Contents

The gamma function | 520 |

Topics in harmonic analysis | 535 |

I | 1 |

The elementary theory of arithmetic functions | 35 |

Principles and first examples of sieve methods | 76 |

I | 108 |

II | 137 |

The Prime Number Theorem | 168 |

Primitive characters and Gauss sums | 282 |

Analytic properties of the zeta function and Lfunctions | 326 |

II | 358 |

Explicit formulæ | 397 |

Conditional estimates | 419 |

Zeros | 452 |

Oscillations of error terms | 463 |

Name index | 544 |

Applications of the Prime Number Theorem | 199 |

Further discussion of the Prime Number Theorem | 244 |

APPENDICES | xi |

Subject index 550 | xix |

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### Common terms and phrases

absolutely convergent Acta Arith Akad Amer analytic arithmetic function Assume RH asymptotic Cambridge character mod character modulo Corollary Deduce denote the number derive Dirichlet characters Dirichlet series error term estimate Euler Euler product exceptional zero Exercise ﬁelds ﬁrst follows formula Fourier half-plane Hence identity implicit constant inequality interval Landau left-hand side Legendre symbol Lemma li(x log log logarithmic derivatives logn logx logy London Math lower bound main term Mellin transform Mertens mod q modulo q multiplicative non-negative non-principal character number of primes obtain Perron's formula polynomial positive integer power series Prime Number Theorem primitive character modulo primitive quadratic character Proc proof of Theorem quadratic character real number residue classes Riemann Riemann Hypothesis Riemann zeta function right-hand side Show sieve square-free Suppose taking Tauberian Thales Verlag Theory Uber uniformly upper bound zeta function