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Average order of an arithmetic function: Difference between revisions

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The error bound for the case \alpha = 1 is not O(x), but O(x log x) as proved in several of the references of the article.
→Examples: An average order is never unique; the average value, if it exists, is unique
 
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* An average order of {{math|''φ''(''n'')}}, [[Euler's totient function]] of {{math|''n''}}, is {{math|6''n''{{thinsp|/}}π<sup>2</sup>}};
* 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==
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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 {{math|1=gcd(''a'', ''b'') = ''d'' > 1}}, then writing ''a'' = ''da''<sup>2</sup>, ''b'' = ''db''<sup>2</sup> one observes that the point (''a''<sup>2</sup>, ''b''<sup>2</sup>) 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.
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