Möbius Function

The Möbius function is a number theoretic function defined by

(1)

so  indicates that  is squarefree (Havil 2003, p. 208). The first few values of  are therefore 1, , , 0, , 1, , 0, 0, 1, , 0, ... (OEIS A008683). Similarly, the first few values of  for , 2, ... are 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, ... (OEIS A008966).

The function was introduced by Möbius (1832), and the notation  was first used by Mertens (1874). However, Gauss considered the Möbius function more than 30 years before Möbius, writing "The sum of all primitive roots [of a prime number ] is either  (when  is divisible by a square), or  (mod ) (when  is the product of unequal prime numbers; if the number of these is even the sign is positive but if the number is odd, the sign is negative)" (Gauss 1801, Pegg 2003).

The Möbius function is implemented in the Wolfram Language as MoebiusMu[n].

The summatory function of the Möbius function

(2)

is called the Mertens function.

The following table gives the first few values of  for , 0, and 1. The values of the first  integers are plotted above on a  grid, where values of  with  are shown in red,  are shown in black, and  are shown in blue. Clear patterns emerge where multiples of numbers each share one or more repeated factor.

	OEIS	values of
	A030059	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, ...
0	A013929	4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, ...
1	A030229	1, 6, 10, 14, 15, 21, 22, 26, ...

The Möbius function has generating functions

(3)

for  (Nagell 1951, p. 130). This product follows by taking one over the Euler product and expanding the terms to obtain

			(4)
			(5)
			(6)
			(7)
			(8)

(Derbyshire 2004, pp. 245-249).

An additional generating function is given by

(9)

for . It also obeys the infinite sums

			(10)
			(11)
			(12)
			(13)
			(14)

(OEIS A082020, A088245, and A088245; Havil 2003, p. 208), as well as the divisor sum

(15)

where  is the number of distinct prime factors of  (Hardy and Wright 1979, p. 235).

 also satisfies the infinite product

(16)

for  (Bellman 1943; Buck 1944;, Pólya and Szegö 1976, p. 126; Robbins 1999). Equation (◇) is as "deep" as the prime number theorem (Landau 1909, pp. 567-574; Landau 1911; Hardy 1999, p. 24).

The Möbius function is multiplicative,

(17)

and satisfies

(18)

where  is the Kronecker delta, as well as

(19)

where  is the number of divisors (i.e., divisor function of order zero; Nagell 1951, p. 281).

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "The Möbius Function." §24.3.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 826, 1972.

Bellman, R. "Problem 4072." Amer. Math. Monthly 50, 124-125, 1943.

Buck, R. C. "Solution to Problem 4072." Amer. Math. Monthly 51, 410, 1944.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, pp. 245-250, 2004.

Gauss, C. F. §81 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. Translated by A. A. Clarke. New Haven, CT: Yale University Press, 1965.

Hardy, G. H. "A Note on the Möbius Function." §4.9 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 64-65, 1999.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford: Clarendon Press, p. 236, 1979.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.

Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen. Leipzig, Germany: Teubner, 1909.

Landau, E. Prac. Matematyczno-Fizycznych 21, 97-177, 1910.

Landau, E. Wiener Sitzungsber. 120, 973-988, 1911.

Mertens, F. "Über einige asymptotische Gesetze der Zahlentheorie." J. reine angew. Math. 77, 46-62, 1874.

Miller, J. "Earliest Uses of Symbols of Number Theory." https://members.aol.com/jeff570/nth.html.

Möbius, A. F. "Über eine besondere Art von Umkehrung der Reihen." J. reine angew. Math. 9, 105-123, 1832.

Nagell, T. Introduction to Number Theory. New York: Wiley, p. 27, 1951.

Pegg, E. Jr. "Math Games: The Möbius Function (and Squarefree Numbers)." Nov. 3, 2003. https://www.maa.org/editorial/mathgames/mathgames_11_03_03.html.

Pólya, G. and Szegö, G. Problems and Theorems in Analysis, Vol. 2. New York: Springer-Verlag, 1976.

Robbins, N. "Some Identities Connecting Partition Functions to Other Number Theoretic Functions." Rocky Mtn. J. Math. 29, 335-345, 1999.

Rota, G.-C. "On the Foundations of Combinatorial Theory I. Theory of Möbius Functions." Z. für Wahrscheinlichkeitsth. 2, 340-368, 1964.

Séroul, R. "The Moebius Function." §2.12 and 8.5 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 19-21 and 167-169, 2000.

Sloane, N. J. A. Sequences A008683, A008966, A013929, A030059, A030229, A082020, A88245, and A88246 in "The On-Line Encyclopedia of Integer Sequences."

Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 7-8 and 223-225, 1991.

Wilf, H. Generatingfunctionology, 2nd ed. New York: Academic Press, p. 61, 1994.