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Function equal to the product of its values on coprime factors
{\displaystyle b} are coprime. An arithmetic function is said to be completely multiplicative (or totally multiplicative) if f ( 1 ) = 1 {\displaystyle f(1)=1}
Multiplicative_function
Arithmetic function
theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions
Completely multiplicative function
Completely_multiplicative_function
Number which when multiplied by x equals 1
is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a b {\displaystyle {\tfrac {a}{b}}}
Multiplicative_inverse
Number of integers coprime to and less than n
1 ) = 1 {\displaystyle \gcd(1,1)=1} . Euler's totient function is a multiplicative function, meaning that if two numbers m {\displaystyle m} and n {\displaystyle
Euler's_totient_function
Identity obeyed by many special functions related to the gamma function
polygamma function is the logarithmic derivative of the gamma function, and thus, the multiplication theorem becomes additive, instead of multiplicative: k m
Multiplication_theorem
Multiplicative function in number theory
The Möbius function μ ( n ) {\displaystyle \mu (n)} is a multiplicative function in number theory introduced by the German mathematician August Ferdinand
Möbius_function
Mathematical operation on arithmetical functions
Dirichlet convolution of two multiplicative functions is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse
Dirichlet_convolution
Function whose domain is the positive integers
the zeta function gives the Möbius inversion formula: f(n)=∑d∣nμ(nd)g(d). If f is multiplicative, then so is g. If f is completely multiplicative, then g
Arithmetic_function
Function on an integer n which is log(p) if n equals p^k and zero otherwise
an important arithmetic function that is neither multiplicative nor additive. The von Mangoldt function, denoted by Λ ( n ) {\displaystyle \Lambda (n)}
Von_Mangoldt_function
Arithmetical function
In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by ψ ( n ) = n ∏ p | n ( 1 + 1 p ) , {\displaystyle
Dedekind_psi_function
Function studied by Ramanujan
n} are coprime (that is, τ ( n ) {\displaystyle \tau (n)} is a multiplicative function) τ ( p r + 1 ) = τ ( p ) τ ( p r ) − p 11 τ ( p r − 1 ) {\displaystyle
Ramanujan_tau_function
Topics referred to by the same term
Multiplicative may refer to: Multiplication Multiplicative function Multiplicative group Multiplicative identity Multiplicative inverse Multiplicative
Multiplicative
Product of the prime factors of an integer
(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
Radical_of_an_integer
Association of one output to each input
compute the zeros of the function, the values where the function is defined but not its multiplicative inverse. Similarly, a function of a complex variable
Function_(mathematics)
Function that returns its argument unchanged
completely multiplicative function (essentially multiplication by 1), considered in number theory. In a metric space the identity function is trivially
Identity_function
Function that can be written as a sum over prime factors
used in this sense by analogy with totally multiplicative functions. The restriction of the logarithmic function to N . {\displaystyle \mathbb {N} .} The
Additive_function
Arithmetical function
positive integer k {\displaystyle k} , Jordan's totient function J k {\displaystyle J_{k}} is multiplicative and may be evaluated as J k ( n ) = n k ∏ p | n (
Jordan's_totient_function
Concept in modular arithmetic
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
Modular multiplicative inverse
Modular_multiplicative_inverse
Largest integer that divides given integers
be used to denote the GCD of multiple arguments. The GCD is a multiplicative function in the following sense: if a1 and a2 are relatively prime, then
Greatest_common_divisor
Function in mathematical number theory
λ function, the reduced totient function, and the least universal exponent function. The order of the multiplicative group of integers modulo n is φ(n)
Carmichael_function
Analytic function in mathematics
(1992). The Riemann Zeta-Function. Berlin, DE: W. de Gruyter. Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative Number Theory. I. Classical
Riemann_zeta_function
Integer that divides another integer
total number of positive divisors of n {\displaystyle n} is a multiplicative function d ( n ) , {\displaystyle d(n),} meaning that when two numbers m
Divisor
Performing order of mathematical operations
is replaced with multiplication by the reciprocal (multiplicative inverse), then the associative and commutative laws of multiplication allow the factors
Order_of_operations
Arithmetic function
(a)+\Omega (b)} , then λ ( n ) {\displaystyle \lambda (n)} is completely multiplicative. Since 1 {\displaystyle 1} has no prime factors, Ω ( 1 ) = 0 {\displaystyle
Liouville_function
Function in number theory
may not be a quadratic residue mod p. The Legendre symbol is a multiplicative function. The Legendre symbol was introduced by Adrien-Marie Legendre in
Legendre_symbol
Mapping arbitrary data to fixed-size values
possibly faster hash function. Selected divisors or multipliers in the division and multiplicative schemes may make more uniform hash functions if the keys are
Hash_function
Function in number theory
In number theory, the unit function is a completely multiplicative function on the positive integers defined as: ε ( n ) = { 1 , if n = 1 0 , if n ≠
Unit_function
Problem in number theory on equal totients
mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ ( n ) {\displaystyle \varphi (n)}
Carmichael's totient function conjecture
Carmichael's_totient_function_conjecture
Topics referred to by the same term
(polynomials), the greatest common divisor of the coefficients is a multiplicative function Gauss's lemma (number theory), condition under which an integer
Gauss's_lemma
equal; this is sometimes called the uniqueness theorem: given multiplicative functions f {\displaystyle f} and g {\displaystyle g} , one has f = g {\displaystyle
Bell_series
Arithmetical operation
generalizations See Multiplication in group theory, above, and multiplicative group, which for example includes matrix multiplication. A very general, and
Multiplication
Formal power series
generating function is especially useful when an is a multiplicative function, in which case it has an Euler product expression in terms of the function's Bell
Generating_function
Certain type of divisor of an integer
unitary divisors of n are multiplicative functions of n that are not completely multiplicative. The Dirichlet generating function is ζ ( s ) ζ ( s − k )
Unitary_divisor
Analytic function that does not satisfy a polynomial equation
addition, subtraction, multiplication, and division (without the need of taking limits). This is in contrast to an algebraic function. The most familiar transcendental
Transcendental_function
Concept in modular arithmetic
to n, the multiplicative order of a modulo n is the smallest positive integer k such that ak ≡ 1 (mod n). In other words, the multiplicative order of a
Multiplicative_order
Arithmetic operation
invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted 1
Exponentiation
The lowest common divisor is a term mistakenly used to refer to: Lowest common denominator, the lowest common multiple of the denominators of a set of
Lowest_common_divisor
Topics referred to by the same term
generalized multiplicative function, in number theory Multiply (website), e-commerce website based in Jakarta, Indonesia Multiplication of money, the
Multiplication (disambiguation)
Multiplication_(disambiguation)
Mathematician
collaboration, covers the theory of the Riemann zeta function, random multiplicative functions, S-unit equations, smooth numbers, the large sieve, and
Adam_Harper
Mathematical operation in linear algebra
as matrix multiplication (up to a multiplicative constant), the computational complexity of matrix multiplication appears throughout numerical linear
Matrix_multiplication
The greatest common multiple is a term mistakenly used to refer to: Least common denominator, the lowest common multiple of the denominators of a set of
Greatest_common_multiple
Extension of the factorial function
normalization of the gamma function is the integral of the additive character e − x {\displaystyle e^{-x}} against the multiplicative character x z {\displaystyle
Gamma_function
Mathematical concept
the multiplicative inverse of f(x) and has nothing to do with the inverse function of f. The notation f⟨−1⟩ might be used for the inverse function to avoid
Inverse_function
Number of partitions of an integer
an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal
Partition function (number theory)
Partition_function_(number_theory)
Type of mathematical expression
polynomial function; that is, the evaluation consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and
Polynomial
Unsolved problem in mathematics
Unsolved problem in mathematics Can the totient function of a composite number n {\displaystyle n} divide n − 1 {\displaystyle n-1} ? More unsolved problems
Lehmer's_totient_problem
Finnish mathematician
distribution of multiplicative functions over short intervals of numbers; for instance, she showed that the values of the Möbius function are evenly divided
Kaisa_Matomäki
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
ν {\displaystyle \nu } function is closely related to the ξ {\displaystyle \xi } function which is the multiplicative function defined by ξ ( p n ) =
Lemniscate_constant
Act of placing two elements side by side
times x {\displaystyle x} . It is also used for scalar multiplication, matrix multiplication, function composition, and logical and. In numeral systems, juxtaposition
Juxtaposition
Property of a number
is the smallest number of multiplicative persistence 3. In base 10, there is thought to be no number with a multiplicative persistence greater than 11;
Persistence_of_a_number
Mathematical symbol
programming language to denote the sign function. The lower-case Latin letter x is sometimes used in place of the multiplication sign. This is considered incorrect
Multiplication_sign
Topics referred to by the same term
satisfies some predicate Möbius function μ(n), a multiplicative function in number theory and combinatorics Membership function (mathematics), μA, the m of
MU
Exploring properties of the integers with complex analysis
about the Dirichlet series. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series (or a product of simpler
Analytic_number_theory
cryptanalysis Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula Divisor function Liouville
List_of_number_theory_topics
Mathematical function associated to algebraic varieties
in the case of multiplicative reduction ap is ±1 depending on whether E has split (plus sign) or non-split (minus sign) multiplicative reduction at p
Hasse–Weil_zeta_function
Way to write a number as a product of other numbers
pointwise. Although the study of multiplicative partitions has been ongoing since at least 1923, the name "multiplicative partition" appears to have been
Multiplicative_partition
Mathematical function, inverse of an exponential function
discrete logarithm in the multiplicative group of non-zero elements of a finite field. Further logarithm-like inverse functions include the double logarithm ln(ln(x))
Logarithm
Mathematical formula
single-digit remains, which is called the multiplicative digital root of n {\displaystyle n} . The multiplicative digital root for the first few positive
Multiplicative_digital_root
Mathematical series
if there exists an inverse function such that the Dirichlet convolution of f with its inverse yields the multiplicative identity ∑ d | n f ( d ) f −
Dirichlet_series
Algebraic structure with addition, multiplication, and division
+ (−a) = 0. Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a−1 or 1/a, called the multiplicative inverse of a
Field_(mathematics)
Functions such that f(–x) equals f(x) or –f(x)
In mathematics, an even function is a real function such that f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} for every x {\displaystyle x} in its domain
Even_and_odd_functions
Evaluation of a function on its argument
In mathematics, function application (or evaluation) is the act of taking a function and an input from its domain to obtain the corresponding value from
Function_application
Functions of an angle
and arccos, etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids
Trigonometric_functions
Types of special mathematical functions
defined similarly to the gamma function but with different or "incomplete" integral limits. The usual gamma function is defined as an integral from zero
Incomplete_gamma_function
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Algorithm to multiply two numbers
product of 9 and 3. In prehistoric time, quarter square multiplication involved floor function; that some sources attribute to Babylonian mathematics (2000–1600
Multiplication_algorithm
Computational operation
Inverse: [(−a mod n) + (a mod n)] mod n = 0. b−1 mod n denotes the modular multiplicative inverse, which is defined if and only if b and n are relatively prime
Modulo
Infinite products of functions indexed by primes
would later become known as the Riemann zeta function. In general, if a is a bounded multiplicative function, then the Dirichlet series ∑ n = 1 ∞ a ( n
Euler_product
Function with a multiplicative scaling behaviour
mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by
Homogeneous_function
Group of units of the ring of integers modulo n
the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
Ways to estimate the size of sifted sets of integers
where g ( d ) {\displaystyle g(d)} is a density, meaning a multiplicative function such that g ( 1 ) = 1 , 0 ≤ g ( p ) < 1 p ∈ P {\displaystyle g(1)=1
Sieve_theory
Generalization of additive and multiplicative inverses
-1}} is not commonly used for function composition, since 1 f {\textstyle {\frac {1}{f}}} can be used for the multiplicative inverse. If x and y are invertible
Inverse_element
Topics referred to by the same term
x, the multiplicative inverse (or reciprocal) of the trigonometric function cosine (see above for ambiguity)[citation needed] Inverse function sec−1 (disambiguation)
Cos-1
Mathematics award
multiplicative functions." Maksym Radziwill – "For fundamental breakthroughs in the understanding of local correlations of values of multiplicative functions
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Theorem in mathematics
inverse function. The inverse function is also continuously differentiable, and the inverse function rule expresses its derivative as the multiplicative inverse
Inverse_function_theorem
Algebraic structure in linear algebra
w, and called the sum of these two vectors. The binary function, called scalar multiplication, assigns to any scalar a in F and any vector v in V another
Vector_space
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
operation: Additive function: preserves the addition operation: f (x + y) = f (x) + f (y). Multiplicative function: preserves the multiplication operation: f (xy)
List_of_types_of_functions
Number with a real and an imaginary part
an additive inverse −z and every nonzero complex number z has a multiplicative inverse 1/z; the complex numbers form a field, much as the rational
Complex_number
Mathematical table
columns for multiplication by 1, the multiplicative identity, which satisfies a × 1 = a. The traditional rote learning of multiplication was based on
Multiplication_table
Integer
can be proved using the distributive law and the axiom that 1 is the multiplicative identity: x + (−1) ⋅ x = 1 ⋅ x + (−1) ⋅ x = (1 + (−1)) ⋅ x = 0 ⋅ x =
−1
Potential counterexample to the generalized Riemann hypothesis
arithmetic function χ : Z → C {\textstyle \chi \colon \mathbb {Z} \to \mathbb {C} } satisfying the following properties: Completely multiplicative: χ ( m
Siegel_zero
In mathematics, invariant of square matrices
and all other entries are zero. A determinant function that satisfies the definition by multiplicative properties satisfy the definition by elementary
Determinant
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution
Softmax_function
Mathematical function, denoted exp(x) or e^x
the additive identity 0 to the multiplicative identity 1. The same equation is satisfied by other continuous functions f ( x ) = b x {\displaystyle f(x)=b^{x}}
Exponential_function
Specific element of an algebraic structure
respect to multiplication is called a multiplicative identity (often denoted as 1). These need not be ordinary addition and multiplication—as the underlying
Identity_element
Real function with secant line between points above the graph itself
function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function
Convex_function
Counting technique in combinatorics
The Dirichlet hyperbola method re-expresses a sum of a multiplicative function f(n) by selecting a suitable Dirichlet convolution f=g∗h, recognizing
Inclusion–exclusion_principle
Topics referred to by the same term
1/tan(x) = cot(x) or cotangent of x, the multiplicative inverse (or reciprocal) of the trigonometric function tangent (see above for ambiguity) tan x−1
Tan-1
Computation modulo a fixed integer
possessing a multiplicative inverse. They form an abelian group under multiplication; its order is φ(m), where φ is Euler's totient function. In pure mathematics
Modular_arithmetic
Inverse functions of sin, cos, tan, etc.
than function composition, and therefore may result in confusion between notation for the reciprocal (multiplicative inverse) and inverse function. The
Inverse trigonometric functions
Inverse_trigonometric_functions
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Estimate size of sifted sets
A_{d}\right\vert ={\frac {1}{f(d)}}X+R_{d}.} where f is a multiplicative function and X = |A|. Let the function g be obtained from f by Möbius inversion, that
Selberg_sieve
Algebraic operation
scalar multiplication is a function from K × V to V. The result of applying this function to k in K and v in V is denoted kv. Scalar multiplication obeys
Scalar_multiplication
Degree of differentiability of a function or map
In calculus and mathematical analysis, the smoothness of a function describes the number of times it can be differentiated without producing discontinuities
Smoothness
Algorithmic runtime requirements for common math procedures
would imply that the exponent of matrix multiplication is 2. Algorithms for computing transforms of functions (particularly integral transforms) are widely
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Mathematical transform that expresses a function of time as a function of frequency
takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output
Fourier_transform
Quickly growing function
Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not
Ackermann_function
Linear operator acting on modular forms
proved the Hecke relations for the Ramanujan tau function, which implies that it is a multiplicative function, a property conjectured by Ramanujan. The idea
Hecke_operator
Average uncertainty in variable's states
Shannon entropy, but also it used the Liouville function along with averages of modulated multiplicative functions in short intervals. Proving it also broke
Entropy_(information_theory)
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MULTIPLICATIVE FUNCTION
MULTIPLICATIVE FUNCTION
MULTIPLICATIVE FUNCTION
MULTIPLICATIVE FUNCTION
MULTIPLICATIVE FUNCTION
MULTIPLICATIVE FUNCTION
MULTIPLICATIVE FUNCTION
MULTIPLICATIVE FUNCTION
MULTIPLICATIVE FUNCTION
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