Additive function: Difference between revisions

Revision as of 13:14, 1 July 2021

Revision as of 19:37, 23 August 2021

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{{more footnotes|date=February 2013}}

In [[number theory]], an '''{{anchor|definition-additive_function-number_theory}}additive function''' is an [[arithmetic function]] ''f''(''n'') of the positive [[integer]] variable ''n'' such that whenever ''a'' and ''b'' are [[coprime]], the function ofapplied to the product ''ab'' is the sum of the functionsvalues of the function applied to ''a'' and ''b'':<ref name="Erdos1939">Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077746/ online]</ref>

:''f''(''ab'') = ''f''(''a'') + ''f''(''b'').

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== Examples ==

ExampleExamples of arithmetic functions which are completely additive are:

* The restriction of the [[logarithm|logarithmic function]] to '''N'''.

* The '''multiplicity''' of a [[prime number|prime]] factor ''p'' in ''n'', that is the largest exponent ''m'' for which ''p^m'' [[divisor|divides]] ''n''.

* ''a''₀(''n'') -– the sum of primes dividing ''n'' counting multiplicity, sometimes called sopfr(''n''), the potency of ''n'' or the integer logarithm of ''n'' {{OEIS|A001414}}. For example:

::''a''₀(4) = 2 + 2 = 4

::''a''₀(20) = ''a''₀(2² · 5) = 2 + 2 + 5 = 9

::''a''₀(27) = 3 + 3 + 3 = 9

::''a''₀(144) = ''a''₀(2⁴ · 3²) = ''a''₀(2⁴) + ''a''₀(3²) = 8 + 6 = 14

::''a''₀(2,0002000) = ''a''₀(2⁴ · 5³) = ''a''₀(2⁴) + ''a''₀(5³) = 8 + 15 = 23

::''a''₀(2,0032003) = 2003

::''a''₀(54,032,858,972,279) = 1240658

::''a''₀(54,032,858,972,302) = 1780417

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::Ω(27) = Ω(3·3·3) = 3

::Ω(144) = Ω(2⁴ · 3²) = Ω(2⁴) + Ω(3²) = 4 + 2 = 6

::Ω(2,0002000) = Ω(2⁴ · 5³) = Ω(2⁴) + Ω(5³) = 4 + 3 = 7

::Ω(2,0012001) = 3

::Ω(2,0022002) = 4

::Ω(2,0032003) = 1

::Ω(54,032,858,972,279) = 3

::Ω(54,032,858,972,302) = 6

::Ω(20,802,650,704,327,415) = 7

ExampleExamples of arithmetic functions which are additive but not completely additive are:

* ω(''n''), defined as the total number of ''different''distinct [[prime factor#Omega functions|prime factors]] of ''n'' {{OEIS|A001221}}. For example:

::ω(4) = 1

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::ω(27) = ω(3³) = 1

::ω(144) = ω(2⁴ · 3²) = ω(2⁴) + ω(3²) = 1 + 1 = 2

::ω(2,0002000) = ω(2⁴ · 5³) = ω(2⁴) + ω(5³) = 1 + 1 = 2

::ω(2,0012001) = 3

::ω(2,0022002) = 4

::ω(2,0032003) = 1

::ω(54,032,858,972,279) = 3

::ω(54,032,858,972,302) = 5

::ω(20,802,650,704,327,415) = 5

* ''a''₁(''n'') -– the sum of the distinct primes dividing ''n'', sometimes called sopf(''n'') {{OEIS|A008472}}. For example:

::''a''₁(1) = 0

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::''a''₁(27) = 3

::''a''₁(144) = ''a''₁(2⁴ · 3²) = ''a''₁(2⁴) + ''a''₁(3²) = 2 + 3 = 5

::''a''₁(2,0002000) = ''a''₁(2⁴ · 5³) = ''a''₁(2⁴) + ''a''₁(5³) = 2 + 5 = 7

::''a''₁(2,0012001) = 55

::''a''₁(2,0022002) = 33

::''a''₁(2,0032003) = 2003

::''a''₁(54,032,858,972,279) = 1238665

::''a''₁(54,032,858,972,302) = 1780410

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:<math>\mathcal{M}_{f^2}(x) = x E^2(x) + O(x D^2(x)).</math>

There is always an absolute constant <math>C_f > 0</math> such that for all [[natural numbersnumber]]s <math>x \geq 1</math>,

:<math>\sum_{n \leq x} |f(n) - E(x)|^2 \leq C_f \cdot x D^2(x).</math>

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Let

:<math> \nu(x; z) := \frac{1}{x} \#\!\left\{n \leq x: \frac{f(n)-A(x)}{B(x)} \leq z\right\}\!.</math>

Suppose that <math>f</math> is an additive function with <math>-1 \leq f(p^{\alpha}) = f(p) \leq 1</math>