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2009-05-06T23:31:28Z

Created page with '==Definition== The '''Dedekind series''' is a generalization of Dirichlet series from the ring of rational integers to the more general case of the [[ring of integers in...'

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==Definition==

The '''Dedekind series''' is a generalization of [[Dirichlet series]] from the [[ring of rational integers]] to the more general case of the [[ring of integers in a number field]].

Suppose <math>K</math> is a [[number field]], i.e., a finite extension of the [[field of rational numbers]] <math>\mathbb{Q}</math>. Suppose <math>\mathcal{O}</math> is the ring of integers in <math>K</math>. Suppose <math>f</math> is a function from the set of nonzero ideals in <math>\mathcal{O}</math> to <math>\mathbb{C}</math>.

The Dedekind series of <math>f</math> is defined as:

<math>s \mapsto \sum_I \frac{f(I)}{(N(I))^s}</math>.

Here, the summation is over all nonzero ideals <math>I</math> of <math>\mathcal{O}</math>, and <math>N(I)</math> denotes the norm of the ideal, which is also equal to the index of <math>I</math> in <math>\mathcal{O}</math> as a subgroup.

Dedekind series - Revision history

Vipul: Created page with '==Definition== The '''Dedekind series''' is a generalization of Dirichlet series from the ring of rational integers to the more general case of the [[ring of integers in...'