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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
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
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
Function whose domain is the positive integers
arithmetic function a is completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n; completely multiplicative if a(1) = 1 and a(mn) =
Arithmetic_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
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
Idk is the completely multiplicative function Id k ( n ) = n k {\displaystyle \operatorname {Id} _{k}(n)=n^{k}} . The divisor function σ k {\displaystyle
Bell_series
Function that returns its argument unchanged
is a completely multiplicative function (essentially multiplication by 1), considered in number theory. In a metric space the identity function is trivially
Identity_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
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
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
Function in number theory
{a}{p}}\right)=\left({\frac {b}{p}}\right).} The Legendre symbol is a completely multiplicative function of its top argument: ( a b p ) = ( a p ) ( b p ) . {\displaystyle
Legendre_symbol
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
Complex-valued arithmetic function
\chi (ab)=\chi (a)\chi (b);} that is, χ {\displaystyle \chi } is completely multiplicative. 2. χ ( a ) = 0 ⟺ gcd ( a , m ) > 1 {\displaystyle \chi (a)=0\iff
Dirichlet_character
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)
Type of mathematical function
L-function has a simple pole at s=1. Otherwise, the L-function is entire. Since a Dirichlet character χ is completely multiplicative, its L-function can
Dirichlet_L-function
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
Product of the first "n" prime numbers
where φ {\displaystyle \varphi } is the Euler totient function. Any completely multiplicative function is defined by its values at primorials, since it is
Primorial
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
Generalization of the Legendre symbol in number theory
the top or bottom argument is fixed, the Jacobi symbol is a completely multiplicative function in the remaining argument: ( a b n ) = ( a n ) ( b n ) , {\displaystyle
Jacobi_symbol
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
Mathematical series
{f(n)\log(n)}{n^{s}}}} assuming the right hand side converges. For a completely multiplicative function f(n), and assuming the series converges for Re(s) > σ0, then
Dirichlet_series
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
Mathematical description of quantum state
every possible square integrable function. The state of such a particle is completely described by its wave function, Ψ ( x , t ) , {\displaystyle \Psi
Wave_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
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
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
Root of a quadratic polynomial with a unit leading coefficient
(this is false if D > 0 {\textstyle D>0} ). The norm is a completely multiplicative function, which means that the norm of a product of quadratic integers
Quadratic_integer
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
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)
Objects that generalize functions
analysis, distributions are a type of generalized function that makes it possible to differentiate functions whose derivatives do not exist in the classical
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Norm on a vector space of matrices
} can be rescaled to be sub-multiplicative; in some books, the terminology matrix norm is reserved for sub-multiplicative norms. A matrix norm is called
Matrix_norm
Mathematical relation making a non-equal comparison
additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function. If the inequality is
Inequality_(mathematics)
Algorithm to multiply matrices
recursively, using the scalar multiplication c11 = a11b11 as its base case. The complexity of this algorithm as a function of n is given by the recurrence
Matrix multiplication algorithm
Matrix_multiplication_algorithm
Algorithm for generating pseudo-randomized numbers
that specify the generator. If c = 0, the generator is often called a multiplicative congruential generator (MCG), or Lehmer RNG. If c ≠ 0, the method is
Linear_congruential_generator
Calculator designed to calculate problems in science, engineering, and mathematics
subtraction, multiplication, division) and advanced (trigonometric, hyperbolic, etc.) mathematical operations and functions. They have completely replaced
Scientific_calculator
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
Algorithm for public-key cryptography
The possibility of using Euler totient function results also from Lagrange's theorem applied to the multiplicative group of integers modulo pq. Thus any
RSA_cryptosystem
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
Arithmetic operation
nonzero numbers have a multiplicative inverse. In these cases, a division by x may be computed as the product by the multiplicative inverse of x. This approach
Division_(mathematics)
Function computed by two parties that emulates a random oracle
An oblivious pseudorandom function (OPRF) is a cryptographic function, similar to a keyed-hash function, but with the distinction that in an OPRF two
Oblivious pseudorandom function
Oblivious_pseudorandom_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
Authenticated encryption mode for block ciphers
constructed by feeding blocks of data into the GHASH function and encrypting the result. This GHASH function is defined by GHASH ( H , A , C ) = X m + n +
Galois/Counter_Mode
Output of a dynamic system when given a brief input
transform of a system's output may be determined by the multiplication of the transfer function with the input's Laplace transform in the complex plane
Impulse_response
Type of continuous linear operator
) d y , {\displaystyle (Kf)(x)=\int _{a}^{b}k(x,y)f(y)\,dy,} where the function k {\displaystyle k} is called the integral kernel. Under suitable regularity
Compact_operator
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Unsolved problem in mathematics
the tau function is not completely multiplicative, the sums cannot be written using geometric series like in the case of the Riemann zeta function or Dirichlet
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Mathematical conjecture about zeros of L-functions
\rightarrow \mathbb {C} } of modulus q is an arithmetic function that is: completely multiplicative: χ ( a ⋅ b ) = χ ( a ) ⋅ χ ( b ) {\textstyle \chi (a\cdot
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Axioms for the natural numbers
{\displaystyle S(0)} is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined: S ( 0 ) {\displaystyle
Peano_axioms
Bounded operators with sub-unit norm
form φ ⋅ H∞ where g is an inner function, i.e. such that |φ| = 1 on S1: φ is uniquely determined up to multiplication by a complex number of modulus 1
Contraction_(operator_theory)
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
the result of directly sampling the Dirac delta function. The Kronecker delta forms the multiplicative identity element of an incidence algebra. The Kronecker
Kronecker_delta
Integration kernels for smoothing out sharp features
kernel is one of the functions nowadays called mollifiers. However, since the properties of a linear integral operator are completely determined by its kernel
Mollifier
Meromorphic function
Bernstein's theorem on monotone functions that, for m > 0 and x real and non-negative, (−1)m+1 ψ(m)(x) is a completely monotone function. Setting m = 0 in the above
Polygamma_function
Function spaces generalizing finite-dimensional p norm spaces
function" (or norm), that is: only the zero vector has zero length, the length of the vector is positive homogeneous with respect to multiplication by
Lp_space
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
Set of quantities in probability theory
1112/plms/s2-30.1.199. hdl:2440/15200. Speicher, Roland (1994). "Multiplicative functions on the lattice of non-crossing partitions and free convolution"
Cumulant
Mathematical representation in functional analysis
of continuous functions on the spectrum σ(x) into A such that It maps 1 to the multiplicative identity of A; It maps the identity function on the spectrum
Gelfand_representation
Category of regression analysis
to Nonparametric regression. HyperNiche, software for nonparametric multiplicative regression. Scale-adaptive nonparametric regression (with Matlab software)
Nonparametric_regression
Generalization of the real numbers
fraction, the power function x ∈ N o {\textstyle x\in \mathbb {No} } , x ↦ xy may be composed from multiplication, multiplicative inverse and square root
Surreal_number
Algorithm for fast modular multiplication
Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing fast modular multiplication. It was introduced
Montgomery modular multiplication
Montgomery_modular_multiplication
Branch of mathematics
is invertible only if its determinant has a multiplicative inverse in the ring. Vector spaces are completely characterized by their dimension (up to an
Linear_algebra
Mathematical approximation of a function
of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the
Taylor_series
Set with operations obeying given axioms
which the multiplication operation is commutative. Field: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for
Algebraic_structure
2017 paper conceptualizing the transformer-based deep learning architecture
was the use of multiplicative gating units, in which the outputs of some neurons modulate the outputs of others. These multiplicative units are conceptually
Attention_Is_All_You_Need
Correlation of a signal with a time-shifted copy of itself, as a function of shift
t {\displaystyle t} . Subtracting the mean before multiplication yields the auto-covariance function between times t 1 {\displaystyle t_{1}} and t 2 {\displaystyle
Autocorrelation
Ratio of output to input as a function of frequency
the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design
Frequency_response
Associative array for storing key–value pairs
popular multiplicative hash is claimed to have particularly poor run behavior. K-independent hashing offers a way to prove a certain hash function does not
Hash_table
Algebraic curve in mathematics
ingredient is a function of a complex variable, L, the Hasse–Weil zeta function of E over Q. This function is a variant of the Riemann zeta function and Dirichlet
Elliptic_curve
Number of prime factors of a natural number
main subsection of this article above. To be completely precise, let the odd-indexed summatory function be defined as S odd ( x ) := ∑ n ≤ x ω ( n ) [
Prime_omega_function
Ideal in a ring which has properties similar to prime elements
manifold, R is the ring of smooth real functions on M, and x is a point in M, then the set of all smooth functions f with f (x) = 0 forms a prime ideal
Prime_ideal
Theory that attempts to blend economics and ergodic theory
non-ergodicity in economic processes is a repeated multiplicative coin toss, an instance of the binomial multiplicative process. It demonstrates how an expected-value
Ergodicity_economics
important question in mathematics is whether a space can be completely described by the functions defined on it—that is, by its "observables." The Banach–Stone
Banach–Stone_theorem
Recursive integer sequence
can be completely parenthesized, i.e. the number of ways of associating n applications of a binary operator (as in the matrix chain multiplication problem)
Catalan_number
Theorem in geometry
equivalent to the multiplicative version. To prove the equivalence one direction (from Brunn–Minkowski inequality to its multiplicative form), apply the
Brunn–Minkowski_theorem
Variations in data at specific regular intervals less than a year
component. The multiplicative model can be transformed into an additive model by taking the log of the time series; SA Multiplicative decomposition: Y
Seasonality
Hypercomplex number system
{\displaystyle {e_{i}}^{2}=-1\ } for each point in the diagram completely defines the multiplicative structure of the octonions. Each of the seven lines generates
Octonion
a monoid by adding the identity function. Let M be a monoid and Q be a non-empty set. If there exists a multiplicative operation μ:Q×M→Q (q,m)↦qm=μ(q,m)
Semiautomaton
Operations on ordinals that extend classical arithmetic
standard multiplication. α · 0 = 0 · α = 0, and the zero-product property holds: α · β = 0 implies α = 0 or β = 0. The ordinal 1 is a multiplicative identity
Ordinal_arithmetic
Square of numbers with equal row, column and diagonal totals
some other operation. For example, a multiplicative magic square has a constant product of numbers. A multiplicative magic square can be derived from an
Magic_square
Repeated sum of a number's digits
_{b}(a)\cdot \operatorname {dr} _{b}(c)).} This is a consequence of multiplicative compatibility modulo b − 1 {\displaystyle b-1} . Compatibility with
Digital_root
Quantum algorithm for integer factorization
contained in the multiplicative group of integers modulo N, having a multiplicative inverse modulo N. Thus, a has a multiplicative order r modulo N,
Shor's_algorithm
Computation model defining an abstract machine
initial tape contents; δ:(Q∖F)×Γ⇀Q×Γ×{L,R} is a partial function called the transition function, where L is left shift, R is right shift. If δ is not defined
Turing_machine
Next, K 1 ( F ) = F × , {\displaystyle K_{1}(F)=F^{\times },} the multiplicative group of F. The second K-group of a field is described in terms of generators
K-groups_of_a_field
matrix elements are dense in the space of continuous functions, which determine the group completely. Gelfand–Naimark theorem Representation theory И. М
Gelfand–Raikov_theorem
Vector space equipped with a bilinear product
algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A. Conversely,
Algebra_over_a_field
Multivariate derivative (mathematics)
∇f of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) whose value at a point p gives the
Gradient
CPU instruction to simultaneously read and write a value in memory
write a new value into it simultaneously, either with a completely new value or some function of the previous value. These operations prevent race conditions
Read–modify–write
Notion in measure theory
Supp(μ) are identified with their indicator functions. A character on a unital algebra is a multiplicative linear functional with values in the coefficient
Lifting_theory
Alteration in the nucleotide sequence of a genome
effect, alter the product of a gene, or prevent the gene from functioning properly or completely. Mutations can also occur in non-genic regions. A 2007 study
Mutation
Counts the number of necklaces of n colored beads picked from α available colors
n\,N_{n}\,=\,n*(n\,M_{n})} , since the function f ( n ) = n {\displaystyle f(n)=n} is completely multiplicative. Any two of these imply the third, for
Necklace_polynomial
Open problem on 3x+1 and x/2 functions
odd, triple it and add one. In modular arithmetic notation, define the function f as follows: f(n)={n/2if n≡0(mod2),3n+1if n≡1(mod2). Now form a sequence
Collatz_conjecture
Method of notation of very large integers
operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n
Knuth's_up-arrow_notation
Mathematical function
mathematics, a function of a real variable is a function whose domain is a subset of R {\displaystyle \mathbb {R} } . Many real functions that are often
Function_of_a_real_variable
Method for bounding the errors of numerical computations
rounding and measurement errors in mathematical computation by computing function bounds. Numerical methods involving interval arithmetic can guarantee relatively
Interval_arithmetic
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
Useful mathematical identies of sum of divisors function
for f any arithmetic function and g completely multiplicative where φ ( n ) {\displaystyle \varphi (n)} is Euler's totient function and μ ( n ) {\displaystyle
Divisor_sum_identities
Polynomial whose roots are the eigenvalues of a matrix
The term secular function has been used for what is now called characteristic polynomial (in some literature the term secular function is still used).
Characteristic_polynomial
Extension of evidence theory to continuous variables of interest
Except for normalization constant, the above equation completely determines the normal density function for X. Therefore, M ( X → ) {\displaystyle M({\vec
Linear_belief_function
Topological vector spaces
compact subset K of U such that for every test function ϕ {\displaystyle \phi } whose support is completely outside of K, we have T ( ϕ ) = 0. {\displaystyle
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Field extension generated by a one element
generates L × = L − { 0 } {\displaystyle L^{\times }=L-\{0\}} as a multiplicative group, so that every nonzero element of L is a power of γ, i.e. is produced
Simple_extension
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COMPLETELY MULTIPLICATIVE-FUNCTION
COMPLETELY MULTIPLICATIVE-FUNCTION
COMPLETELY MULTIPLICATIVE-FUNCTION
COMPLETELY MULTIPLICATIVE-FUNCTION
COMPLETELY MULTIPLICATIVE-FUNCTION
COMPLETELY MULTIPLICATIVE-FUNCTION
COMPLETELY MULTIPLICATIVE-FUNCTION
COMPLETELY MULTIPLICATIVE-FUNCTION
COMPLETELY MULTIPLICATIVE-FUNCTION
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