In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the 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". An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n. There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime-counting functions. This article provides links to functions of both classes. Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. An arithmetic function a is completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n; completely multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n; Two whole numbers m and n are called coprime if their greatest common divisor is 1, that is, if there is no prime number that divides both of them. Then an arithmetic function a is additive if a(mn) = a(m) + a(n) for all coprime natural numbers m and n; multiplicative if a(mn) = a(m)a(n) for all coprime natural numbers m and n. In this article, and mean that the sum or product is over all prime numbers: and Similarly, and mean that the sum or product is over all prime powers with strictly positive exponent (so k = 0 is not included): The notations and mean that the sum or product is over all positive divisors of n, including 1 and n. For example, if n = 12, then The notations can be combined: and mean that the sum or product is over all prime divisors of n. For example, if n = 18, then and similarly and mean that the sum or product is over all prime powers dividing n. For example, if n = 24, then The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p1 < p2 < ..

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (31)
MATH-313: Number theory I.b - Analytic number theory
The aim of this course is to present the basic techniques of analytic number theory.
MATH-101(g): Analysis I
Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.
CS-101: Advanced information, computation, communication I
Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a
Show more
Related publications (38)

Semiclassical Estimates for Eigenvalue Means of Laplacians on Spheres

Joachim Stubbe, Luigi Provenzano, Paolo Luzzini, Davide Buoso

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper ...
SPRINGER2023

J-domain protein chaperone circuits in proteostasis and disease

Paolo De Los Rios, Duccio Malinverni, Ruobing Zhang

The J-domain proteins (JDP) form the largest protein family among cellular chaperones. In cooperation with the Hsp70 chaperone system, these co-chaperones orchestrate a plethora of distinct functions, including those that help maintain cellular proteostasi ...
ELSEVIER SCIENCE LONDON2022

Spectral analysis for transmission eigenvalue problems with and without the complementing conditions

Jean Louis-Alexandre Fornerod

The interior transmission eigenvalue problem is a system of partial differential equations equipped with Cauchy data on the boundary: the transmission conditions. This problem appears in the inverse scattering theory for inhomogeneous media when, for some ...
EPFL2022
Show more
Related concepts (28)
Dirichlet series
In mathematics, a Dirichlet series is any series of the form where s is complex, and is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet.
Multiplicative function
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.
Möbius function
The Möbius function μ(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted μ(x). For any positive integer n, define μ(n) as the sum of the primitive nth roots of unity.
Show more

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.