Von Mangoldt function
Overview
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the von Mangoldt function is an arithmetic function
Arithmetic function
In number theory, an arithmetic function is a real or complex valued function ƒ defined on the set of natural numbers In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers In number theory, an arithmetic (or...
named after German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician Hans von Mangoldt
Hans Carl Friedrich von Mangoldt
Hans Carl Friedrich von Mangoldt was a German mathematician who contributed to the solution of the prime number theorem.Von Mangoldt completed his Doctor of Philosophy in 1878 at the University of Berlin, where his advisors were Ernst Kummer and Karl Weierstrass...
.
The von Mangoldt function, conventionally written as Λ(n), is defined as
It is an example of an important arithmetic function that is neither multiplicative
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
nor additive
Additive function
In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive...
.
The von Mangoldt function satisfies the identity
that is, the sum is taken over all integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s d which divide
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
n.
