Moebius Mu function

[Pieter Moree]

Dear seqfans:

> I recently wrote a column for maa.org for the Moebius function.
> http://www.maa.org/editorial/mathgames/mathgames_11_03_03.html
>
> One item here that was new to me, courtesy of Fred W. Helenius,
> was the work Gauss did on this function for his work /Disquisitiones
> Arithmeticae/. There is a nice sequence there, apparently not in OEIS
> yet.
>
> 1, 2, 5, 8, 23, 26, 68, 57, 139, 174, 123, 222, 328, 257, 612,
> 636, 886, 488, 669, 1064, 876, 1105, 1744, 1780, 1552, 2020,
> 1853, 2890, 1962, 2712, 2413, 3536, 4384, 3335, 5364, 3322,
> 3768, 4564, 7683, 7266, 8235, 4344, 8021, 6176, 8274, 6965,
> 5698, 7581, 13167, 8244.
>
> Sums of primitive roots of primes. Gauss proved:
> Sum of primitive roots of prime p=µ(/p/-1) (mod /p/)

An unsolved question regarding this assertion is: does the set
of primes for which \mu(p-1)=1 (say) has a density, d(1) ?

I.e. does \sum_{p\le x: \mu(p-1)=1}1/\pi(x), with \pi(x) the
number of primes p<=x has a limit has x tends to infinity ?

It is a result that goes back to Mirsky that the set of primes p
for which p-1 is squarefree has density A, where A denotes the
Artin constant (A=\prod_q (1-1/(q(q-1)), q running over all primes).
Numerically A=0.3739558136....

Thus the density of primes for which \mu(p-1)=0, d(0), exists and equals
1-A.

Conjecture: d(-1)=d(1)=A/2.

The sum of all primitive roots of a prime p is a totally symmetric
expression in the summands. One can generalize this by considering other
totally symmetric functions in primitive roots, for
example
s_2(p):=sum g_ig_j with i<j or S_k(p):=sum g_i^k, where the sum runs
over all primitive roots mod p.
For these functions similar
(but more complicated) results as Gauss's hold.

Mod p these function behave as if a primitive root mod p equals
a (p-1)th primitive root of unity. Thus connections with
totally symmetric
functions in primitive roots of unity arise. Most famous examples
are Ramanujan sums (related
to S_k's) and coefficients of cyclotomic polynomials (s_k's).

I assigned studying these functions
and their value distribution as a MSc. project of Huib Hommerson
(but also did a lot of work on it myself).
He recently defended his MSc thesis at the University of Amsterdam.

Vide Section 9 of:
P. Moree and H. Hommerson, Value distribution of Ramanujan sums and of
cyclotomic polynomial coefficients, arXiv:math.NT/0307352

or vide:
http://staff.science.uva.nl/~moree/preprints.html

Bests,
Pieter Moree