Encyclopedia > Dirichlet convolution

  Article Content

Dirichlet convolution

The Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory.

See also:

If f and g are two arithmetic functions (i.e. functions from the positive integers to the complex numbers), one defines a new arithmetic function f * g, the Dirichlet convolution of f and g, by

<math>
(f*g)(n) = \sum_{d|n} f(d)g(n/d) </math> where the sum extends over all positive divisors d of n.

Some general properties of this operation include:

  • If both f and g are multiplicative, then so is f * g. (Note however that the convolution of two completely multiplicative functions need not be completely multiplicative.)
  • f * g = g * f (commutativity)
  • (f * g) * h = f * (g * h) (associativity)
  • f * (g + h) = f * g + f * h (distributivity)
  • f * ε = ε * f = f, where ε is the function defined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1.
  • To every multiplicative f there exists a multiplicative g such that f * g = ε.

With addition and Dirichlet convolution, the set of arithmetic functions forms a commutative ring with multiplicative identity ε, the Dirichlet ring. The units of this ring are the arithmetical functions f with f(1) ≠ 0.

Furthermore, the multiplicative functions with convolution form an abelian group with identity element ε. The article on multiplicative functions lists several convolution relations among important multiplicative functions.

If f is an arithmetic function, one defines its L-series by

<math>
L(f,s) = \sum_{n=1}^\infty \frac{f(n)}{n^s} </math> for those complex arguments s for which the series converges (if there are any). The multiplication of L-series is compatible with Dirichlet convolution in the following sense:
<math>
L(f,s) L(g,s) = L(f*g,s) </math> for all s for which the left hand side exists. This is akin to the convolution theorem if one thinks of L-series as a Fourier transform.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
U.S. presidential election, 1804

... Jefferson (W) 162 Democratic-Republican George Clinton (162) Charles C. Pinckney[?] 14 Federalist Rufus King (14) Other elections: 1792, 1796, ...

 
 
 
This page was created in 32.4 ms