In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average".
It is conventional to choose an approximating function
, one has, Generalized identities of the previous form are found here.
This identity often provides a practical way to calculate the mean value in terms of the Riemann zeta function.
We calculate the natural density of these numbers in ℕ, that is, the average value of
is multiplicative, and since it is bounded by 1, its Dirichlet series converges absolutely in the half-plane
Now, if gcd(a, b) = d > 1, then writing a = da2, b = db2 one observes that the point (a2, b2) is on the line segment which joins (0,0) to (a, b) and hence (a, b) is not visible from the origin.
Conversely, it is also easy to see that gcd(a, b) = 1 implies that there is no other integer lattice point in the segment joining (0,0) to (a,b).
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 ℕ.
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 ℕ.
Let h(x) be a function on the set of monic polynomials over Fq.
exists, it is said that h has a mean value (average value) c. Let Fq[X] = A be the ring of polynomials over the finite field Fq.
Similar to the situation in N, every Dirichlet series of a multiplicative function h has a product representation (Euler product):
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,
In a similar way, If f and g are two polynomial arithmetic functions, one defines f * g, the Dirichlet convolution of f and g, by
where the sum extends over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomials whose product is m. The identity
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.
, which is the density of the k-th power free polynomials in Fq[X], in the same fashion as in the integers.
the number of k-th power monic polynomials of degree n, we get
Finally, expand the left-hand side in a geometric series and compare the coefficients on
which resembles the analogous result for the integers:
Expanding the right-hand side into power series we get,
which resembles closely the analogous result for integers
This is because of the very simple nature of the zeta function
The Polynomial von Mangoldt function is defined by:
{\displaystyle \Lambda _{A}(f)={\begin{cases}\log |P|&{\text{if }}f=|P|^{k}{\text{ for some prime monic}}P{\text{ and integer }}k\geq 1,\\0&{\text{otherwise.
Define Euler totient function polynomial analogue,