Average order of an arithmetic function: Difference between revisions

Revision as of 05:23, 1 April 2024

Latest revision as of 08:09, 25 August 2025

(6 intermediate revisions by 5 users not shown)

Line 10:

<math display="block">\lim_{N \to \infty} \frac{1}{N}\sum_{n \le N} f(n)=c</math>

exists, it is said that <math>f</math> has a '''mean value''' ('''average value''') <math>c</math>. If in addition the constant <math>c</math> is not zero, then the constant function <math>g(x)=c</math> is an average order of <math>f</math>.

==Examples==

Line 17:

* An average order of {{math|''φ''(''n'')}}, [[Euler's totient function]] of {{math|''n''}}, is {{math|6''n''{{thinsp|/}}π²}};

* An average order of {{math|''r''(''n'')}}, the number of ways of expressing {{math|''n''}} as a sum of two squares, is {{math|π}};

* TheAn average order of representations of a natural number as a sum of three squares is {{math|4π''n''{{thinsp|/}}3}};

* TheAn average number of decompositions of a natural number into a sum of one or more consecutive prime numbers is {{math|''n'' log2}};

* An average order of {{math|''ω''(''n'')}}, the [[Prime factors|number of distinct prime factors]] of {{math|''n''}}, is {{math|loglog ''n''}};

* An average order of {{math|Ω(''n'')}}, the [[Prime factors|number of prime factors]] of {{math|''n''}}, is {{math|loglog ''n''}};

* The [[prime number theorem]] is equivalent to the statement that the [[von Mangoldt function]] {{math|Λ(''n'')}} has average ordervalue 1;

* AnThe average value of {{math|''μ''(''n'')}}, the [[Möbius function]], is zero; this is again equivalent to the [[prime number theorem]].

==Calculating mean values using Dirichlet series==

Line 35:

===The density of the ''k''^th-power-free integers in {{mathbb|N}}===

For an integer <math>k \geq 1</math> the set <math>Q_k</math> of '''''k''-^th-power-free''' integers is

<math display="block">Q_k :=\{n \in \mathbb{Z}\mid n \text{ is not divisible by } d^k \text{ for any integer } d\ge 2\}.</math>

Line 73:

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 {{math|1=gcd(''a'', ''b'') = ''d'' > 1}}, then writing ''a'' = ''da''², ''b'' = ''db''² one observes that the point (''a''², ''b''²) 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 <math>\frac{\varphi(n)}{n}</math> is the probability of a random point on the square <math>\{(r,s)\in \mathbb{N} : \max(|r|,|s|)=n\}</math> to be visible from the origin.

Line 91:

\sum_{n\le x}\sigma_{\alpha}(n)=

\begin{cases}

\;\;\sum_{n\le x}\sigma_\alpha(n)=\frac{\zeta(\alpha+1)}{\alpha+1}x^{\alpha+1}+O(x^\beta) & \text{if } \alpha>0,\alpha \ne 1, \\

\;\;\sum_{n\le x}\sigma_{1}(n)=\frac{\zeta(2)}{2}x^2+O(x \log x) & \text{if } \alpha=1, \\

\;\;\sum_{n\le x}\sigma_{-1}(n)=\zeta(2)x+O(\log x) & \text{if } \alpha=-1, \\

\;\;\sum_{n\le x}\sigma_\alpha(n)=\zeta(-\alpha+1)x+O(x^{\max(0,1+\alpha)}) & \text{otherwise.}