Dedekind psi function
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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, the Dedekind psi function is the
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 f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime. An arithmetic function ''f''(''n'') i ...
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 multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
, has value 1.) The function was introduced by Richard Dedekind in connection with
modular function In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
s. 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-fr ...
then \psi(n) = \sigma(n), where \sigma(n) is the
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 as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includin ...
. 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 terms of the Riemann zeta function, which is :\sum \frac = \frac. This is also a consequence of the fact that we can write as a
Dirichlet convolution In mathematics, the Dirichlet 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 are two arithmetic fun ...
of \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 In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...
.


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
Dirichlet convolution In mathematics, the Dirichlet 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 are two arithmetic fun ...
of a power and the square of the
Möbius function The Möbius function 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 of ...
, :\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).


References


External links

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See also

* (page 25, equation (1)) * Section 3.13.2 * is ψ2, is ψ3, and {{OEIS2C, A065960 is ψ4 Multiplicative functions