MoebiusMu

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

μ(n) is a nonanalytical function which is defined only for integers.

Symmetries and periodicities

Parity

μ(n) is an even function.

Periodicity

No periodicity

Series representations

Other series representations

Transformations

Multiple arguments

Products, sums, and powers of the direct function

Products of the direct function

Summation

Finite summation

Infinite summation

Ed Pegg

Ed Pegg

Ed Pegg

Ed Pegg

Ed Pegg

Products

Infinite products

S.-.J. Kang, M.-H. Kim: Free Lie Algebras, Generalized Witt Formula, and the Denominator Identity Journal of Algebra 183, 560-594 (1996)

Representations through equivalent functions

With related functions

Zeros

Theorems

The Möbius inversion theorem

F(n)==(∑_(d|n))f(d)⟺f(n)==(∑_(d|n))F(d)μ((n/d))

or

F(x)==(∑_(k=1)^∞)f((k^α)x),α∈"ℝ",α≠0,x∈"ℝ"⟺f(x)==(∑_(k=1)^∞)F((k^α)x)μ(k).

Generalized Mertens conjecture

For ε>0 and sufficiently large n: |(∑_(k=1)^n)μ(k)|≤(n^((1/2)+ε)).

History

– C. F. Gauss in his Disquisitiones Arithmeticae (1801)
– A. F. Möbius (1831)
– F. Mertens (1874) introduced the notation μ(n) and the name "Möbius function"
– P. L. Chebyshev (1851) and R. Dedekind (1857) found the Möbius inversion theorem

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