# On the Properties of the Möbius Function

Magdalena Jastrzebska; Adam Grabowski

Formalized Mathematics (2006)

- Volume: 14, Issue: 1, page 29-36
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topMagdalena Jastrzebska, and Adam Grabowski. "On the Properties of the Möbius Function." Formalized Mathematics 14.1 (2006): 29-36. <http://eudml.org/doc/266801>.

@article{MagdalenaJastrzebska2006,

abstract = {We formalized some basic properties of the Möbius function which is defined classically as [...] as e.g., its multiplicativity. To enable smooth reasoning about the sum of this number-theoretic function, we introduced an underlying many-sorted set indexed by the set of natural numbers. Its elements are just values of the Möbius function.The second part of the paper is devoted to the notion of the radical of number, i.e. the product of its all prime factors.The formalization (which is very much like the one developed in Isabelle proof assistant connected with Avigad's formal proof of Prime Number Theorem) was done according to the book [13].},

author = {Magdalena Jastrzebska, Adam Grabowski},

journal = {Formalized Mathematics},

language = {eng},

number = {1},

pages = {29-36},

title = {On the Properties of the Möbius Function},

url = {http://eudml.org/doc/266801},

volume = {14},

year = {2006},

}

TY - JOUR

AU - Magdalena Jastrzebska

AU - Adam Grabowski

TI - On the Properties of the Möbius Function

JO - Formalized Mathematics

PY - 2006

VL - 14

IS - 1

SP - 29

EP - 36

AB - We formalized some basic properties of the Möbius function which is defined classically as [...] as e.g., its multiplicativity. To enable smooth reasoning about the sum of this number-theoretic function, we introduced an underlying many-sorted set indexed by the set of natural numbers. Its elements are just values of the Möbius function.The second part of the paper is devoted to the notion of the radical of number, i.e. the product of its all prime factors.The formalization (which is very much like the one developed in Isabelle proof assistant connected with Avigad's formal proof of Prime Number Theorem) was done according to the book [13].

LA - eng

UR - http://eudml.org/doc/266801

ER -

## References

top- [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
- [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
- [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [4] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.
- [5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [6] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.
- [7] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
- [8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [9] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [10] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
- [11] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
- [12] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
- [13] G.H. Hardy and E.M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 1980. Zbl0020.29201
- [14] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4):573-577, 1997.
- [15] Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179-186, 2004.
- [16] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471-475, 1990.
- [17] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
- [18] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.
- [19] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.
- [20] Library Committee of the Association of Mizar Users. Binary operations on numbers. To appear in Formalized Mathematics.
- [21] Piotr Rudnicki. Little Bezout theorem (factor theorem). Formalized Mathematics, 12(1):49-58, 2004.
- [22] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.
- [23] Christoph Schwarzweller and Andrzej Trybulec. The evaluation of multivariate polynomials. Formalized Mathematics, 9(2):331-338, 2001.
- [24] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.
- [25] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
- [26] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.
- [27] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.
- [28] Andrzej Trybulec. Many-sorted sets. Formalized Mathematics, 4(1):15-22, 1993.
- [29] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.
- [30] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 1990.
- [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- [33] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.