Function equal to the product of its values on coprime factors
Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and
whenever a and b are coprime.
An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b, even when they are not coprime.
and 26 Related for: Multiplicative function information
a and b are coprime. An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds...
convolution of two multiplicativefunctions is multiplicative, the Dirichlet convolution of two completely multiplicativefunctions need not be completely...
mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity...
polygamma function is the logarithmic derivative of the gamma function, and thus, the multiplication theorem becomes additive, instead of multiplicative: k m...
Dirichlet convolution of two multiplicativefunctions is again multiplicative, and every not constantly zero multiplicativefunction has a Dirichlet inverse...
f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative. There...
τ(n): τ(mn) = τ(m)τ(n) if gcd(m,n) = 1 (meaning that τ(n) is a multiplicativefunction) τ(pr + 1) = τ(p)τ(pr) − p11 τ(pr − 1) for p prime and r > 0. |τ(p)|...
the space. The identity function on the positive integers is a completely multiplicativefunction (essentially multiplication by 1), considered in number...
analogy with totally multiplicativefunctions. If f is a completely additive function then f(1) = 0. Every completely additive function is additive, but not...
total number of positive divisors of n {\displaystyle n} is a multiplicativefunction d ( n ) , {\displaystyle d(n),} meaning that when two numbers m...
translates into a single integer multiplication and right-shift making it one of the fastest hash functions to compute. Multiplicative hashing is susceptible to...
an important arithmetic function that is neither multiplicative nor additive. The von Mangoldt function, denoted by Λ(n), is defined as Λ ( n ) = { log...
be used to denote the GCD of multiple arguments. The GCD is a multiplicativefunction in the following sense: if a1 and a2 are relatively prime, then...
\ 1{\pmod {n}}} . In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the...
} The accuracy of the result justifies an attempt to derive a multiplicativefunction for that average, for example, y ∼ x + 2.67. {\displaystyle y\sim...
(504)=2\cdot 3\cdot 7=42} The function r a d {\displaystyle \mathrm {rad} } is multiplicative (but not completely multiplicative). The radical of any integer...
generating function is especially useful when an is a multiplicativefunction, in which case it has an Euler product expression in terms of the function's Bell...
generalizations See Multiplication in group theory, above, and multiplicative group, which for example includes matrix multiplication. A very general, and...
solution, i.e., when it exists, a modular multiplicative inverse is unique: If b and b' are both modular multiplicative inverses of a respect to the modulus...
about the Dirichlet series. Thus a common method for estimating a multiplicativefunction is to express it as a Dirichlet series (or a product of simpler...
(n)=(-1)^{\Omega (n)}.} (sequence A008836 in the OEIS). λ is completely multiplicative since Ω(n) is completely additive, i.e.: Ω(ab) = Ω(a) + Ω(b). Since...
called unity. Consequently, if f ( x ) {\displaystyle f(x)} is a multiplicativefunction, then f ( 1 ) {\displaystyle f(1)} must be equal to 1. This distinctive...
Möbius function, an important multiplicativefunction in number theory and combinatorics Möbius transform, transform involving the Möbius function Möbius...
In number theory, the Legendre symbol is a multiplicativefunction with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p:...
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