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Everything about Arithmetic function
Function whose domain is the positive integers
In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Arithmetic functionA 5-minute orientation
- Skim a recent paper: Effective arithmetic optimization algorithm with probabilistic search strategy for function optimization problemsInfluential work since 2022
- Pick up a classic book: Advanced Functional AnalysisThe most-reprinted book on the subject
- Read the landmark paper: Arithmetic of quantum entropy functionCited 99 times
- Go sideways: Multiplicative functionA closely related topic
- Go sideways: Ramanujan's sumA closely related topic
📖 Definitions
Noun Any function that is defined for all positive integers, and has values that are either real or complex.
🗂️ The subject at a glance
- Multiplicative and additive functions
- Notation
- Ω(n), ω(n), νp(n) – prime power decomposition
- Multiplicative functions
- Completely multiplicative functions
- Additive functions
- Completely additive functions
- Neither multiplicative nor additive
- Summation functions
- Dirichlet convolution
- Relations among the functions
- First 100 values of some arithmetic functions
🏆 Landmark papers (most cited)
Arithmetic of quantum entropy function
Arithmetic properties of the partition function
On an additive arithmetic function
The divisor function over arithmetic progressions
Arithmetic of function field units
On an Arithmetic Function (II)
The Γ-function in the arithmetic of function fields
Ultra-arithmetic I: Function data types
🆕 Influential since 2022
Effective arithmetic optimization algorithm with probabilistic search strategy for function optimization problems
The Race Arithmetic of the Global Lung Function Initiative Global Reference Equations
An arithmetic term for the factorial function
On sequence spaces defined by arithmetic function and Hausdorff measure of noncompactness
Arithmetic properties of the Herglotz-Zagier-Novikov function
📚 Books worth reading
Advanced Functional AnalysisEberhard Malkowsky, Vladimir Rakočević · 2019- 📕Architecture of MathematicsSimon Serovajsky · 2020
Introduction to analytic number theoryTom M. Apostol · 1976
Generalized Trigonometric and Hyperbolic FunctionsRonald E. Mickens · 2019
Arithmetic Differential Operators Over The Padic IntegersClaire C. Ralph · 2012
The arithmetic of elliptic curvesJoseph H. Silverman · 1985- 📕Elementa arithmeticæ numerosæ et speciosæEdward Wells · 1698
Arithmetical functionsWolfgang Schwarz · 1994
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Multiplicative function In number theory, a multiplicative function is an arithmetic function f of a positive integer n with the property that f(1)=1 and f(ab)=f(a)f(b) whenever…
- Ramanujan's sum In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula cq(n)=∑1≤a≤q(a…
- Average order of an arithmetic function In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average". Let…
- Euler's totient function This article uses technical mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a natural…
- Dirichlet convolution In mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory…
- Prime omega function In number theory, the prime omega functions ω(n) and Ω(n) count the number of prime factors of a natural number n. The number of distinct prime factors…
- Divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to…
- Von Mangoldt function In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important…