The prime nu function is the arithmetic function on the positive integers given by
 |
(1)
|
when the prime factorization of 
is
 |
(2)
|
where the 
are distinct primes and the 
are positive integers.
Thus 
counts the number of distinct prime factors
of 
,
without multiplicity, and 
. It is implemented in the Wolfram
Language as PrimeNu[n].
The first few values of 
for 
, 2, ... are 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2,
1, 1, 2, 1, 2, ... (OEIS A001221, Kac 1959,
Abramowitz and Stegun 1972). For example, 
because 
has the two distinct
prime factors 2 and 3. The sequence is the inverse Möbius
transform of 
,
where 
is the characteristic function of the
prime numbers (Sloane and Plouffe 1995, p. 22).
The function is strongly additive,
 |
(3)
|
when 
and 
are relatively prime, while 
for every prime 
and positive
integer 
.
Some authors denote the function by 
(Hardy and Wright 1979, p. 354). Care is needed,
since 
can instead denote the number of all divisors of 
(Ore 1988, p. 86).
A sum involving 
is
 |
(4)
|
for 
(Hardy and Wright 1979, p. 255).
For a positive integer 
, let
 |
(5)
|
An asymptotic series for this mean is
 |
(6)
|
where 
is the Mertens constant and the 
are Stieltjes constants
(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003). The corresponding variance
over the integers 1, 2, ..., 
has expansion
 |
(7)
|
where
 |  |  |
(8)
|
 |  |  |
(9)
|
(Finch 2003, p. 7), and
 |
(10)
|
(OEIS A085548) is the prime zeta function 
(Finch 2003). The coefficients are
 |  |  |
(11)
| ||
 |  |  |
(12)
| ||
 |  |  |
(13)
| ||
 |  | ![-gamma_1-(gamma-1)[gamma+2sum_(k=1)^(infty)(lnp_k)/(p_k(p_k-1))]](/images/equations/PrimeNuFunction/Inline45.svg) |  |  |
(14)
|
 |  |  |
(15)
|
Equivalently, the sums occurring here have values
 |  |  |
(16)
|
 |  |  |
(17)
|
(Finch 2003).
The average order of 
is 
(Hardy 1999, p. 51). If 
is a primorial, then
 |
(18)
|
(Hardy and Wright 1979, p. 355). Its summatory function satisfies
 |
(19)
|
where 
is the Mertens constant (Hardy and Ramanujan
1917; Hardy and Wright 1979, p. 355; Hardy 1999, p. 57). The first few
values of the summatory function for 
, 3, ... are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17,
19, 20, 21, ... (OEIS A013939). In addition,
![sum_(k=2)^n[omega(k)]^2=n(lnlnn)^2+O(nlnlnn)](/images/equations/PrimeNuFunction/NumberedEquation11.svg) |
(20)
|
(Hardy and Wright 1979, p. 357).
The first few squarefree numbers 
with an odd number of distinct
prime factors are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43,
47, ... (OEIS A030059; Hardy 1999, p. 64;
Ramanujan 2000, pp. xxiv and 21). They satisfy
![sum_(n=1)^infty1/(u_n^s)=1/2[(zeta(s))/(zeta(2s))-1/(zeta(s))]](/images/equations/PrimeNuFunction/NumberedEquation12.svg) |
(21)
|
for 
(Hardy 1999, pp. 64-65). If 
counts the 
, then
 |
(22)
|
(Hardy 1999, pp. 64-65).
." Dept. Statistics Tech. Report 96, Stanford, CA: Stanford
University, 1976.Diaconis, P. "G. H. Hardy and Probability???"
Bull. London Math. Soc. 34, 385-402, 2002.Finch, S. "Two
Asymptotic Series." Dec. 10, 2003. 
." Quart. J. Math. 48, 76-92, 1917.Hardy,
G. H. and Wright, E. M. §§22.10-22.11 in