Average order of an arithmetic function


In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average".
Let be an arithmetic function. We say that an average order of is if
as tends to infinity.
It is conventional to choose an approximating function that is continuous and monotone. But even so an average order is of course not unique.
In cases where the limit
exists, it is said that has a mean value .

Examples

In case is of the form
for some arithmetic function, one has,
Generalizations of the previous identity are found here. This identity often provides a practical way to calculate the mean value in terms of the Riemann zeta function. This is illustrated in the following example.

The density of the k-th power free integers in

For an integer the set of k-th-power-free integers is
We calculate the natural density of these numbers in, that is, the average value of indicator function|, denoted by, in terms of the zeta function.
The function is multiplicative, and since it is bounded by 1, its Dirichlet series converges absolutely in the half-plane, and there has Euler product
By the Möbius inversion formula, we get
where stands for the Möbius function. Equivalently,
where
and hence,
By comparing the coefficients, we get
Using, we get
We conclude that,
where for this we used the relation
which follows from the Möbius inversion formula.
In particular, the density of the square-free integers is.

Visibility of lattice points

We say that two lattice points are visible from one another if there is no lattice point on the open line segment joining them.
Now, if gcd = d > 1, then writing a = da2, b = db2 one observes that the point
is on the line segment which joins to and hence is not visible from the origin. Thus is visible from the origin implies that = 1. Conversely, it is also easy to see that gcd = 1 implies that there is no other integer lattice point in the segment joining to.
Thus, is visible from if and only if gcd = 1.
Notice that is the probability of a random point on the square to be visible from the origin.
Thus, one can show that the natural density of the points which are visible from the origin is given by the average,
is also the natural density of the square-free numbers in. In fact, this is not a coincidence. Consider the k-dimensional lattice,. The natural density of the points which are visible from the origin is, which is also the natural density of the k-th free integers in.

Divisor functions

Consider the generalization of :
The following are true:
where.

Better average order

This notion is best discussed through an example. From
and
we have the asymptotic relation
which suggests that the function is a better choice of average order for than simply.

Mean values over

Definition

Let h be a function on the set of monic polynomials over Fq. For we define
This is the mean value of h on the set of monic polynomials of degree n. We say that g is an average order of h if
as n tends to infinity.
In cases where the limit,
exists, it is said that h has a mean value c.

Zeta function and Dirichlet series in

Let =A be the ring of polynomials over the finite field.
Let h be a polynomial arithmetic function. Its corresponding Dirichlet series define to be
where for, set if, and otherwise.
The polynomial zeta function is then
Similar to the situation in, every Dirichlet series of a multiplicative function h has a product representation :
Where the product runs over all monic irreducible polynomials P.
For example, the product representation of the zeta function is as for the integers:.
Unlike the classical zeta function, is a simple rational function:
In a similar way, If ƒ and g are two polynomial arithmetic functions, one defines ƒ * g, the Dirichlet convolution of ƒ and g, by
where the sum extends over all monic divisors d of m, or equivalently over all pairs of monic polynomials whose product is m. The identity still holds. Thus, like in the elementary theory, the polynomial Dirichlet series and the zeta function has a connection with the notion of mean values in the context of polynomials. The following examples illustrate it.

Examples

The density of the ''k''-th power free polynomials in

Define to be 1 if is k-th power free and 0 otherwise.
We calculate the average value of, which is the density of the k-th power free polynomials in, in the same fashion as in the integers.
By multiplicativity of :
Denote the number of k-th power monic polynomials of degree n, we get
Making the substitution we get:
Finally, expand the left-hand side in a geometric series and compare the coefficients on on both sides, to conclude that
Hence,
And since it doesn't depend on n this is also the mean value of.

Polynomial Divisor functions

In, we define
We will compute for.
First, notice that
where and.
Therefore,
Substitute we get,
Finally we get that,
Notice that
Thus, if we set then the above result reads
which resembles the analogous result for the integers:

Number of divisors

Let be the number of monic divisors of f and let be the sum of over all monics of degree n.
where.
Expanding the right-hand side into power series we get,
Substitute the above equation becomes:
Not much is known about the error term for the integers, while in the polynomials case, there is no error term!
This is because of the very simple nature of the zeta function, and that it has NO zeros.

Polynomial von Mangoldt function

The Polynomial von Mangoldt function is defined by:
Where the logarithm is taken on the basis of q.
Proposition. The mean value of is exactly 1.
Proof.
Let m be a monic polynomial, and let be the prime decomposition of m.
We have,
Hence,
and we get that,
Now,
Thus,
We got that:
Now,
Hence,
and by dividing by we get that,

Polynomial Euler totient function

Define Euler totient function polynomial analogue,, to be the number of elements in the group. We have,