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 (average value) . If in addition the constant is not zero, then the constant function is an average order of .

Examples

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Calculating mean values using Dirichlet series

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In case   is of the form   for some arithmetic function  , one has,

  (1)

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

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For an integer   the set   of kth-power-free integers is  

We calculate the natural density of these numbers in ℕ, that is, the average value of  , 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 (1), 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

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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(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. 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.

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 ℕ. 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 ℕ.

Divisor functions

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Consider the generalization of  :  

The following are true:   where  .

Better average order

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

Mean values over Fq[x]

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Definition

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Let h(x) be a function on the set of monic polynomials over Fq. For   we define  

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 if   as n tends to infinity.

In cases where the limit,   exists, it is said that h has a mean value (average value) c.

Zeta function and Dirichlet series in Fq[X]

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Let Fq[X] = A be the ring of polynomials over the finite field Fq.

Let h be a polynomial arithmetic function (i.e. a function on set of monic polynomials over A). Its corresponding Dirichlet series define to be   where for  , set   if  , and   otherwise.

The polynomial zeta function is then  

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,   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, 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   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

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The density of the k-th power free polynomials in Fq[X]

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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 Fq[X], 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

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In Fq[X], we define  

We will compute   for  .

First, notice that   where   and  .

Therefore,  

Substitute   we get,  and by Cauchy product 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

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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:   which resembles closely the analogous result for integers  , where   is Euler constant.

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

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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

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

See also

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References

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  • Hardy, G. H.; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001. pp. 347–360
  • Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics. Vol. 46. Cambridge University Press. pp. 36–55. ISBN 0-521-41261-7. Zbl 0831.11001.
  • Tom M. Apostol (1976), Introduction to Analytic Number Theory, Springer Undergraduate Texts in Mathematics, ISBN 0-387-90163-9
  • Michael Rosen (2000), Number Theory in Function Fields, Springer Graduate Texts In Mathematics, ISBN 0-387-95335-3
  • Hugh L. Montgomery; Robert C. Vaughan (2006), Multiplicative Number Theory, Cambridge University Press, ISBN 978-0521849036
  • Michael Baakea; Robert V. Moodyb; Peter A.B. Pleasantsc (2000), Diffraction from visible lattice points and kth power free integers, Discrete Mathematics- Journal