## Abstract analytic number theoryNorth-Holland Mathematical Library, Volume 12: Abstract Analytic Number Theory focuses on the approaches, methodologies, and principles of the abstract analytic number theory. The publication first deals with arithmetical semigroups, arithmetical functions, and enumeration problems. Discussions focus on special functions and additive arithmetical semigroups, enumeration and zeta functions in special cases, infinite sums and products, double series and products, integral domains and arithmetical semigroups, and categories satisfying theorems of the Krull-Schmidt type. The text then ponders on semigroups satisfying Axiom A, asymptotic enumeration and "statistical" properties of arithmetical functions, and abstract prime number theorem. Topics include asymptotic properties of prime-divisor functions, maximum and minimum orders of magnitude of certain functions, asymptotic enumeration in certain categories, distribution functions of prime-independent functions, and approximate average values of special arithmetical functions. The manuscript takes a look at arithmetical formations, additive arithmetical semigroups, and Fourier analysis of arithmetical functions, including Fourier theory of almost even functions, additive abstract prime number theorem, asymptotic average values and densities, and average values of arithmetical functions over a class. The book is a vital reference for researchers interested in the abstract analytic number theory. |

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### Contents

1 | |

9 | |

ARITHMETICAL SEMIGROUPS WITH ANALYTICAL PROPERTIES OF CLASSICAL TYPE | 73 |

ANALYTICAL PROPERTIES OF OTHER ARITHMETICAL SYSTEMS | 217 |

SOME UNSOLVED QUESTIONS | 287 |

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### Common terms and phrases

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