number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, the Dedekind psi function is the

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 a and b are coprime. An arithmetic function is said to be completely multiplicative (o ...

on the positive integers defined by :

\psi(n) = n \prod_\left(1+\frac\right),

where the product is taken over all primes

p

dividing

n.

(By convention,

\psi(1)

, which is the

empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...

, has value 1.) The function was introduced by

Richard Dedekind Julius Wilhelm Richard Dedekind (; ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. H ...

in connection with modular functions. The value of

\psi(n)

for the first few integers

n

is: :1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... . The function

\psi(n)

is greater than

n

for all

n

greater than 1, and is even for all

n

greater than 2. If

n

is a

square-free number In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square- ...

then

\psi(n) = \sigma(n)

, where

\sigma(n)

is the sum-of-divisors function. The

\psi

function can also be defined by setting

\psi(p^n) = (p+1)p^

for powers of any prime

p

, and then extending the definition to all integers by multiplicativity. This also leads to a proof of the

generating function In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...

in terms of the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

, which is :

\sum \frac = \frac.

This is also a consequence of the fact that we can write as a

Dirichlet convolution In mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet. Definition If f , g : \mathbb\to\mathbb ...

\psi= \mathrm * , \mu,

. There is an additive definition of the psi function as well. Quoting from Dickson,

R. DedekindJournal für die reine und angewandte Mathematik, vol. 83, 1877, p. 288. Cf. H. Weber, Elliptische Functionen, 1901, 244-5; ed. 2, 1008 (Algebra III), 234-5 proved that, if $n$ is decomposed in every way into a product $ab$ and if $e$ is the g.c.d. of $a, b$ then : $\sum_ (a/e) \varphi(e) = n \prod_\left(1+\frac\right)$ where $a$ ranges over all divisors of $n$ and $p$ over the prime divisors of $n$ and $\varphi$ is the totient function.

Higher orders

The generalization to higher orders via ratios of Jordan's totient is :

\psi_k(n)=\frac

with Dirichlet series :

\sum_\frac = \frac

. It is also the

of a power and the square of the

Möbius function The Möbius function \mu(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 m ...

, :

\psi_k(n) = n^k * \mu^2(n)

. If :

\epsilon_2 = 1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0\ldots

is the

characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function \mathbf_A\colon X \to \, which for a given subset ''A'' of ''X'', has value 1 at points ...

of the squares, another Dirichlet convolution leads to the generalized σ-function, :

\epsilon_2(n) * \psi_k(n) = \sigma_k(n)

Higher orders

References

External links

See also