## Abstract Analytic Number Theory"This book is well-written and the bibliography excellent," declared Mathematical Reviews of John Knopfmacher's innovative study. The three-part treatment applies classical analytic number theory to a wide variety of mathematical subjects not usually treated in an arithmetical way. The first part deals with arithmetical semigroups and algebraic enumeration problems; Part Two addresses arithmetical semigroups with analytical properties of classical type; and the final part explores analytical properties of other arithmetical systems.Because of its careful treatment of fundamental concepts and theorems, this text is accessible to readers with a moderate mathematical background, i.e., three years of university-level mathematics. An extensive bibliography is provided, and each chapter includes a selection of references to relevant research papers or books. The book concludes with an appendix that offers several unsolved questions, with interesting proposals for further development. |

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### Contents

Introduction | 1 |

ARITHMETICAL SEMIGROUPS | 9 |

Arithmetical functions | 23 |

Copyright | |

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### Common terms and phrases

absolutely convergent abstract prime number additive arithmetical semigroup algebraic integers algebraic number field analytic function analytic number theory arith arithmetical category arithmetical functions arithmetical semigroup satisfying asymptotic density asymptotic mean-value Chapter classical coefficients complex numbers conclusion consider constant coprime Corollary cyclotomic deduce defined degree mapping denote the total Dir(G discussion distribution function divisor divisor function domain elements aŁG elements of G Euclidean domain Euler product formula example finite abelian groups follows formation structure func function on G Hence homomorphism implies interesting isomorphism classes Knopfmacher Lemma Let G denote log log Math metical modules multiplicative function NG(x norm number of isomorphism p-rings particular PIM-function polynomial positive integers power series prime ideals prime number theorem Proof properties Proposition pseudo-convergent Ramanujan sums real-valued semigroup G semigroup satisfying Axiom sequence shows subset suppose Theorem 3.1 tion topological total number uniformly convergent zeta function