The Möbius Function (and squarefree numbers)

**Ed Pegg Jr.**, November 3, 2003

In 1831, August Ferdinand Möbius put numbers into 3 bins, as a new type of function. His procedure is represented by the Greek letter mu (µ), and is called the Möbius function in his honor.

In the "**zero**" bin, Möbius placed multiples
of square numbers (other than 1). {4 8 9 12 16 18 20 24 25 27 28 32
36 40 44 45 48 49 50 52 54 56 60 63 64 68} -- A013929.
So, µ(48) = 0, µ(49) = 0, and µ(50) = 0, since each
is divisible by a square number.

Already, we have something amazing. The probability that a number
isn't in the "zero" bin just happens to be 6/π^{2}.
Out of the first 100000 numbers, this probability predicts 39207
numbers with µ(*n*) = 0. The actual figure is 39206.
Within that range, the *n*(1-6/π^{2})
guess can be 14 too low, at 80157, or 15 too high, at 47524. A proof
of the 6/π^{2} density can be found at
planetmath.org.

Figure 1. Differences between the actual and predicted density of
square-divisible numbers.

You might notice three numbers in a row in the above list, {48, 49, 50}. It's a nice problem to find the first occurrence of 4 in a row. 22020 is the start of 6 square-divisible numbers in a row. See OEIS entry A045882 for an astounding extension.

In the "**negative one**" bin, Möbius placed any
number that factored into an odd number of distinct primes. {2 3 5 7
11 13 17 19 23 29 30 31 37 41 42 43 47 53 59 61 66 67 70} --
A030059.
So, µ(29) = -1, µ(30) = -1, and µ(31) = -1. Every
fourth number is divisible by 4, so there won't be more than three in
a row in this bin.

The primes are in this bin, along with numbers like 66 (2×3×11),
665 (5×7×19), 6654 (2×3×1109), ...,
66549954999999999 (3×11×61×2238013×14772071)
and 665499549999999999 (3×2063×107529415091291). The
probability that a number falls in the negative one bin is 3/π^{2}.
*P*(µ(*n*) = -1) = 3/π^{2},
in the more standard mathematical notation. The prediction isn't
nearly as accurate, as seen in the plot of differences between the
real and expected values.

Figure 2. Differences between the real and predicted density of µ(*n*) = -1.

In the "**positive one**" bin, Möbius put all
the numbers that factor into an even number of distinct primes. For
completeness, he put 1 into this bin. {1 6 10 14 15 21 22 26 33 34 35
38 39 46 51 55 57 58 62 65 69} --
A030229.

Numbers like 6 (2×3), 69 (3×23), 699 (3×233), 6999, ..., and 69999069090909090990909090 are all in this bin. As a puzzle, there are three distinct digits that can be arranged to make product of two distinct primes. In fact, all six arrangements give the product of two primes. Can you find the smallest of these numbers?

The Möbius
function, µ(*n*), is strongly related to the
Zeta
function ζ(*s*).

Figure 3. Identities of the Riemann Zeta function.

The Möbius function has many amazing identities. Is it possible to figure out µ(googolplex +1)? We know a lot of factors of this number, 316912650057057350374175801344000001, for example. A list of all the factors of googolplex +1 would have more digits than atoms in the universe, so a complete factorization seems impossible. Is there some way of identifying whether a number has an odd or even number of distinct prime factors without factoring?

Perhaps. For ζ(*s*),
consider *s* = 14.1347251417347 I + 1/2. This is (
approximately)
the first nontrivial zero of the
Riemann
Zeta function. ζ(*s*)
= 0, for that value. 1/ζ(s)
= ∞. For these special zeroes on the critical line (the first
half trillion nontrivial ζ_{0}'s
have been computed), the sign of *n*^ζ_{0}
and µ(*n*) are strongly connected. The function
*sgn*(*Re*(*n*^{(14.1347251417347i+1/2)})) can be used to
make a guess at µ(*n*).

Figure 4. Cumulative accuracy of the ζ_{0} guessing method

In the first 100000 tries, the ζ_{0}
idea is correct a thousand instances more than it is wrong. It's not
completely useless. Through this method, a wild guess at the Möbius
value of googolplex+1 is possible, without a complete factorization.

With a bit of number sleuthing, I found that
Table[Sign[Re[1/n^(14.1347251417347 I + 1/2)]] , {n, 1,
2000}] changes signs at points Table[Round[2.17698
1.248896^k], {k, 1, 30}]. Using logarithms, 2.17698
1.248896^k = 10^(10^100) + 1 can be solved. At this too-low
level of accuracy, *k* is closest to an even number, so I can
guess that googolplex+1 has an even number of factors. It would be
nice to redo this calculation with hundreds of digits of accuracy on
all levels, over many ζ_{0}'s. With that, a calculation with a 52%
chance of being correct might be possible. Flip a coin for a 50% chance of correctness. For
smaller numbers, the ζ_{0} method
might accurately distinguish whether 109!+1 has 2 or 3 factors.
Various unfactored composites can be found at the
Factorization
Results page.

For n = 1 to 20, µ(*n*) = {1, -1, -1, 0, -1, 1, -1, 0,
0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0} --
A008683.
The cumulative sum is thus {1, 0, -1, -1, -2, -1, -2, -2, -2, -1, -2,
-2, -3, -2, -1, -1, -2, -2, -3, -3} --
A002321.
This is known as
Mertens
Function, or *M*(*x*). In 1885, Stieltjes claimed to
have a proof that |*M*(*x*)/*x*^{½}| <
1 for all x. In 1897, Merten said the same thing, and now gets all
the credit. The first time |*M*(*x*)/*x*^{½}|
exceeds ½ is at *M*(7725030629) = 43947.

Figure 5. The bounded Mertens function, |*M*(*x*)/*x*^{½}|.

It would seem that the only way to calculate *M*(*x*) is
to sum up the µ(*n*) values for all *n*<*x*,
and the only way to find µ(*n*) is to factor *n*.
However, there is another method, found by E. C. Titchmarsh, which
involves the earlier mentioned solutions for ζ(*s*)
= 0. After constructing a
table
of Zeta function zeros, Andrew Odlyzko used the Titchmarsh method
to find a disproof
of Mertens Conjecture. In 1985, he found an example near
*x*=10^{10^64.1442} where |*M*(*x*)/*x*^{½}|
> 1.06. In 1987, J. Pintz showed that another counterexample could
be found somewhere under 10^{65}.

Applying the Moebius function to Gaussian integers also works.
These complex numbers can always be factored into non-primes 1, -1,
** i**, or -

Figure 6. The Catalan Constant (~0.9159655941772190150546)

In the image below, the
black pixels represent squarefree Gaussian integers.
Is it possible to reach infinity only by adding 1 or ** i**
for each step? I don't know the answer.

Figure 7. The squarefree maze. Start at 0+0** i**, stay on the
black path, and reach infinity.

All in all, a useful function. Gauss may have used µ(*n*)
at least thirty years earlier. In Article 81 of his *Disquisitiones
Arithmeticae* (1801), he writes (as translated by A.A. Clarke):
"The sum of all primitive roots is either == 0 (when *p*-1 is
divisible by a square), or == +-1 (mod. *p*) (when *p*-1 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)." I'm not quite sure what Gauss meant here. When
I sum the numbers relatively prime to *n*, it is equal to 0, mod *n*.
It does seem like Gauss was close.

**References:**

Finch, S. R. *Mathematical
Constants*. Cambridge, England: Cambridge University Press,
pp. 601-602, 2003.

MacTutor History of Mathematics. "August
Ferdinand Möbius."
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mobius.html.

Odlyzko,
Andrew. "Tables of zeros of the Riemann zeta function."
http://www.dtc.umn.edu/%7Eodlyzko/zeta_tables/.

Odlyzko,
A. M. and te Riele, H. J. J. "Disproof of the Mertens
Conjecture." J. reine angew. Math. 357, 138-160, 1985.
http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf.

PlanetMath.org.
"square-free number."
http://planetmath.org/encyclopedia/SquareFree.html.

Sloane,
N. J. A. Sequences A008683
A013929
A030059
A030229
A045882
in "The On-Line Encyclopedia of Integer Sequences."
http://www.research.att.com/~njas/sequences/.

Trott,
Michael. Catalan, MoebiusMu. http://functions.wolfram.com/.

Weisstein, Eric W. Mertens Function, Möbius function,
Riemann Zeta Function. *Eric Weisstein's World of Mathematics.*
http://mathworld.wolfram.com/.

Zetagrid.net.
"Statistics." http://www.zetagrid.net/.

Many thanks to Fred W. Helenius and Dan Asimov for additional commentary.

*Mathematica*
*Code:*

A notebook for this column is available at the Mathematica Information Center .

Figure 1: ListPlot[FoldList[Plus, 0,
Table[If[MoebiusMu[n] == 0, 1, 0], {n, 1, 100000 - 1}]] - Table[n (1
- 6/Pi^2), {n, 1, 100000}], PlotJoined -> True, AspectRatio ->
1/7];

Figure 2: ListPlot[FoldList[Plus,
0, Table[If[MoebiusMu[n] == -1, 1, 0], {n, 1, 100000 - 1}]] - Table[n
(3/Pi^2), {n, 1, 100000}], PlotJoined -> True, AspectRatio ->
1/7];

Figure 4: ListPlot[FoldList[Plus,
0, Table[Sign[Re[1/n^(14.1347251417347 I + 1/2)]] MoebiusMu[n], {n,
1, 100000}]], PlotJoined -> True, AspectRatio -> 1/7];

Figure
5: ListPlot[Drop[FoldList[Plus, 0,
Table[MoebiusMu[n], {n, 1, 100000}]], 1]/Sqrt[Range[100000]],
PlotJoined -> True, AspectRatio -> 1/7, PlotRange -> {-.5,
.5}];

Comments are welcome. Please send comments to Ed Pegg Jr. at
ed@mathpuzzle.com.

Ed Pegg Jr. is the webmaster for mathpuzzle.com.
He maintains the Infocenter
for Wolfram Research.