Wikipedia / Average order of an arithmetic function
Wikipedia / 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".
be an arithmetic function. We say that an average order tends to infinity.
that is continuous and monotone. But even so an average order is of course not unique.
has a mean value (average value.
Examples
An average order of , the number of divisors of , is ;
An average order of , the sum of divisors of , is ;
An average order of , Euler's totient function of , is ;
An average order of , the number of ways of expressing as a sum of two squares, is ;
An average order of representations of a natural number as a sum of three squares is ;
An average number of decompositions of a natural number into a sum of one or more consecutive prime numbers is ;
An average order of , the number of distinct prime factors of , is ;
An average order of , the number of prime factors of , is ;
The prime number theorem is equivalent to the statement that the von Mangoldt function has average value 1;
The average value of , the Möbius function, is zero; this is again equivalent to the prime number theorem.
Calculating mean values using Dirichlet series
, 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. This is illustrated in the following example.
The density of the kth-power-free integers in N
of kth-power-free
We calculate the natural density of these numbers in , that is, the average value of 1_{Q_k}, in terms of the zeta function.
is multiplicative, and since it is bounded by 1, its Dirichlet series, and there has Euler product
By the Möbius inversion stands for the Möbius function
Using
which follows from the Möbius inversion formula.
In particular, the density of the square-free integers.
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 , 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. Thus (a, b) is visible from the origin implies that (a, b) = 1. 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, (a, b) is visible from (0, 0) if and only if gcd(a, b) = 1.
to be visible from the origin.
is also the natural density of the square-free numbers in . In fact, this is not a coincidence. Consider the k, which is also the natural density of the k-th free integers in .
Divisor functions
.
Better average order
is the Euler–Mascheroni constant.
Mean values over Fq[x]
Definition
Let h(x) be a function on the set of monic polynomials over Fq
This is the mean value (average value) of h on the set of monic polynomials of degree n. We say that g(n) is an average order of h as n tends to infinity.
exists, it is said that h has a mean value (average value) c.
Zeta function and Dirichlet series in Fq[X]
Let be the ring of polynomials over the finite field .
Let h be a polynomial arithmetic function (i.e. a function on set of monic polynomials over A otherwise.
Similar to the situation in , every Dirichlet series of a multiplicative function h where the product runs over all monic irreducible polynomials P.
.
Unlike the classical zeta function is a simple rational 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 where the sum extends over all monic divisors d ofm, or equivalently over all pairs (a, b) of monic polynomials whose product is m 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 Fq[X]
is k-th power free and 0 otherwise.
, which is the density of the k-th power free polynomials in , in the same fashion as in the integers.
the number of k-th power monic polynomials of degree n
Finally, expand the left-hand side in a geometric series
And since it doesn't depend on n.
Polynomial Divisor functions
In
.
.
and by Cauchy product
Number of divisors
be the number of monic divisors of f.
Expanding the right-hand side into power series
is Euler constant.
, and that it has no zeros.
Polynomial von Mangoldt function
The Polynomial von Mangoldt function where the logarithm is taken on the basis of q.
Proposition. is exactly 1.
Proof. Let m be the prime decomposition of m.
Polynomial Euler totient function
Define Euler totient function
See also
References
pp.347–360