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24-537: [REDACTED] Look up additive in Wiktionary, the free dictionary. Additive may refer to: Mathematics [ edit ] Additive function , a function in number theory Additive map , a function that preserves the addition operation Additive set-function see Sigma additivity Additive category , a preadditive category with finite biproducts Additive inverse , an arithmetic concept Additive prime ,

48-667: A b ) = f ( a ) + f ( b ) {\displaystyle f(ab)=f(a)+f(b)} holds for all positive integers a and b , even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f (1) = 0. Every completely additive function is additive, but not vice versa. Examples of arithmetic functions which are completely additive are: Examples of arithmetic functions which are additive but not completely additive are: From any additive function f ( n ) {\displaystyle f(n)} it

72-526: A ) = a . Then f ( bc ) = ( bc ) = b c = f ( b ) f ( c ), and f (1) = 1 = 1. The Liouville function is a non-trivial example of a completely multiplicative function as are Dirichlet characters , the Jacobi symbol and the Legendre symbol . A completely multiplicative function is completely determined by its values at the prime numbers, a consequence of the fundamental theorem of arithmetic . Thus, if n

96-499: A function whose domain is the natural numbers ), such that f (1) = 1 and f ( ab ) = f ( a ) f ( b ) holds for all positive integers a and b . In logic notation: f ( 1 ) = 1 {\displaystyle f(1)=1} and ∀ a , b ∈ domain ( f ) , f ( a b ) = f ( a ) f ( b ) {\displaystyle \forall a,b\in {\text{domain}}(f),f(ab)=f(a)f(b)} . Without

120-471: A prime if the sum its is a number which is also prime. Science [ edit ] Additive color , as opposed to subtractive color Additive model , a statistical regression model Additive synthesis , an audio synthesis technique Additive genetic effects Additive quantity, a physical quantity that is additive for subsystems; see Intensive and extensive properties Engineering [ edit ] Feed additive Gasoline additive ,

144-527: A substance used to improve the performance of a fuel, lower emissions or clean the engine Oil additive , a substance used to improve the performance of a lubricant Weakly additive , the quality of preferences in some logistics problems Polymer additive Pit additive , a material aiming to reduce fecal sludge build-up and control odor in pit latrines, septic tanks and wastewater treatment plants Biodegradable additives Other uses [ edit ] Additive case  [ et ] , one of

168-399: Is g ( n ) = 2 f ( n ) . {\displaystyle g(n)=2^{f(n)}.} Likewise if f ( n ) {\displaystyle f(n)} is completely additive, then g ( n ) = 2 f ( n ) {\displaystyle g(n)=2^{f(n)}} is completely multiplicative. More generally, we could consider

192-663: Is a product of powers of distinct primes, say n = p q ..., then f ( n ) = f ( p ) f ( q ) ... While the Dirichlet convolution of two multiplicative functions is multiplicative, the Dirichlet convolution of two completely multiplicative functions need not be completely multiplicative. Arithmetic functions which can be written as the Dirichlet convolution of two completely multiplicative functions are said to be quadratics or specially multiplicative multiplicative functions. They are rational arithmetic functions of order (2, 0) and obey

216-401: Is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions . Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article. A completely multiplicative function (or totally multiplicative function) is an arithmetic function (that is,

240-755: Is an additive function with − 1 ≤ f ( p α ) = f ( p ) ≤ 1 {\displaystyle -1\leq f(p^{\alpha })=f(p)\leq 1} such that as x → ∞ {\displaystyle x\rightarrow \infty } , B ( x ) = ∑ p ≤ x f 2 ( p ) / p → ∞ . {\displaystyle B(x)=\sum _{p\leq x}f^{2}(p)/p\rightarrow \infty .} Then ν ( x ; z ) ∼ G ( z ) {\displaystyle \nu (x;z)\sim G(z)} where G ( z ) {\displaystyle G(z)}

264-1620: Is given exactly as M f ( x ) = ∑ p α ≤ x f ( p α ) ( ⌊ x p α ⌋ − ⌊ x p α + 1 ⌋ ) . {\displaystyle {\mathcal {M}}_{f}(x)=\sum _{p^{\alpha }\leq x}f(p^{\alpha })\left(\left\lfloor {\frac {x}{p^{\alpha }}}\right\rfloor -\left\lfloor {\frac {x}{p^{\alpha +1}}}\right\rfloor \right).} The summatory functions over f {\displaystyle f} can be expanded as M f ( x ) = x E ( x ) + O ( x ⋅ D ( x ) ) {\displaystyle {\mathcal {M}}_{f}(x)=xE(x)+O({\sqrt {x}}\cdot D(x))} where E ( x ) = ∑ p α ≤ x f ( p α ) p − α ( 1 − p − 1 ) D 2 ( x ) = ∑ p α ≤ x | f ( p α ) | 2 p − α . {\displaystyle {\begin{aligned}E(x)&=\sum _{p^{\alpha }\leq x}f(p^{\alpha })p^{-\alpha }(1-p^{-1})\\D^{2}(x)&=\sum _{p^{\alpha }\leq x}|f(p^{\alpha })|^{2}p^{-\alpha }.\end{aligned}}} The average of

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288-463: Is possible to create a related multiplicative function g ( n ) , {\displaystyle g(n),} which is a function with the property that whenever a {\displaystyle a} and b {\displaystyle b} are coprime then: g ( a b ) = g ( a ) × g ( b ) . {\displaystyle g(ab)=g(a)\times g(b).} One such example

312-469: Is that for any completely multiplicative function f one has f ∗ f = τ ⋅ f {\displaystyle f*f=\tau \cdot f} which can be deduced from the above by putting both g = h = 1 {\displaystyle g=h=1} , where 1 ( n ) = 1 {\displaystyle 1(n)=1} is the constant function . Here τ {\displaystyle \tau }

336-451: Is the Gaussian distribution function G ( z ) = 1 2 π ∫ − ∞ z e − t 2 / 2 d t . {\displaystyle G(z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-t^{2}/2}dt.} Examples of this result related to the prime omega function and

360-573: Is the Möbius function . Completely multiplicative functions also satisfy a distributive law. If f is completely multiplicative then f ⋅ ( g ∗ h ) = ( f ⋅ g ) ∗ ( f ⋅ h ) {\displaystyle f\cdot (g*h)=(f\cdot g)*(f\cdot h)} where * represents the Dirichlet product and ⋅ {\displaystyle \cdot } represents pointwise multiplication . One consequence of this

384-468: The Busche-Ramanujan identity. There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function f is multiplicative then it is completely multiplicative if and only if its Dirichlet inverse is μ ⋅ f {\displaystyle \mu \cdot f} where μ {\displaystyle \mu }

408-878: The following way: f ( 1 ) = f ( 1 ⋅ 1 ) ⟺ f ( 1 ) = f ( 1 ) f ( 1 ) ⟺ f ( 1 ) = f ( 1 ) 2 ⟺ f ( 1 ) 2 − f ( 1 ) = 0 ⟺ f ( 1 ) ( f ( 1 ) − 1 ) = 0 ⟺ f ( 1 ) = 0 ∨ f ( 1 ) = 1. {\displaystyle {\begin{aligned}f(1)=f(1\cdot 1)&\iff f(1)=f(1)f(1)\\&\iff f(1)=f(1)^{2}\\&\iff f(1)^{2}-f(1)=0\\&\iff f(1)\left(f(1)-1\right)=0\\&\iff f(1)=0\lor f(1)=1.\end{aligned}}} The definition above can be rephrased using

432-1321: The function f 2 {\displaystyle f^{2}} is also expressed by these functions as M f 2 ( x ) = x E 2 ( x ) + O ( x D 2 ( x ) ) . {\displaystyle {\mathcal {M}}_{f^{2}}(x)=xE^{2}(x)+O(xD^{2}(x)).} There is always an absolute constant C f > 0 {\displaystyle C_{f}>0} such that for all natural numbers x ≥ 1 {\displaystyle x\geq 1} , ∑ n ≤ x | f ( n ) − E ( x ) | 2 ≤ C f ⋅ x D 2 ( x ) . {\displaystyle \sum _{n\leq x}|f(n)-E(x)|^{2}\leq C_{f}\cdot xD^{2}(x).} Let ν ( x ; z ) := 1 x # { n ≤ x : f ( n ) − A ( x ) B ( x ) ≤ z } . {\displaystyle \nu (x;z):={\frac {1}{x}}\#\!\left\{n\leq x:{\frac {f(n)-A(x)}{B(x)}}\leq z\right\}\!.} Suppose that f {\displaystyle f}

456-565: The function g ( n ) = c f ( n ) {\displaystyle g(n)=c^{f(n)}} , where c {\displaystyle c} is a nonzero real constant. Given an additive function f {\displaystyle f} , let its summatory function be defined by M f ( x ) := ∑ n ≤ x f ( n ) {\textstyle {\mathcal {M}}_{f}(x):=\sum _{n\leq x}f(n)} . The average of f {\displaystyle f}

480-460: The grammatical cases in Estonian Food additive , any substance added to food to improve flavor, appearance, shelf life, etc. Additive rhythm , a larger period of time constructed from smaller ones Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Additive . If an internal link led you here, you may wish to change

504-450: The language of algebra: A completely multiplicative function is a homomorphism from the monoid ( Z + , ⋅ ) {\displaystyle (\mathbb {Z} ^{+},\cdot )} (that is, the positive integers under multiplication) to some other monoid. The easiest example of a completely multiplicative function is a monomial with leading coefficient 1: For any particular positive integer n , define f (

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528-454: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Additive&oldid=1238224996 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Additive function An additive function f ( n ) is said to be completely additive if f (

552-1322: The numbers of prime divisors of shifted primes include the following for fixed z ∈ R {\displaystyle z\in \mathbb {R} } where the relations hold for x ≫ 1 {\displaystyle x\gg 1} : # { n ≤ x : ω ( n ) − log ⁡ log ⁡ x ≤ z ( log ⁡ log ⁡ x ) 1 / 2 } ∼ x G ( z ) , {\displaystyle \#\{n\leq x:\omega (n)-\log \log x\leq z(\log \log x)^{1/2}\}\sim xG(z),} # { p ≤ x : ω ( p + 1 ) − log ⁡ log ⁡ x ≤ z ( log ⁡ log ⁡ x ) 1 / 2 } ∼ π ( x ) G ( z ) . {\displaystyle \#\{p\leq x:\omega (p+1)-\log \log x\leq z(\log \log x)^{1/2}\}\sim \pi (x)G(z).} Totally multiplicative In number theory , functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions . A weaker condition

576-476: The requirement that f (1) = 1, one could still have f (1) = 0, but then f ( a ) = 0 for all positive integers a , so this is not a very strong restriction. If one did not fix f ( 1 ) = 1 {\displaystyle f(1)=1} , one can see that both 0 {\displaystyle 0} and 1 {\displaystyle 1} are possibilities for the value of f ( 1 ) {\displaystyle f(1)} in

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