Definition and Basic Properties
Let K be an algebraic number field. Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series
where I ranges through the non-zero ideals of the ring of integers OK of K and NK/Q(I) denotes the absolute norm of I (which is equal to both the index of I in OK or equivalently the cardinality of quotient ring OK / I). This sum converges absolutely for all complex numbers s with real part Re(s) > 1. In the case K = Q, this definition reduces to that of the Riemann zeta function.
Read more about this topic: Dedekind Zeta Function
Famous quotes containing the words definition, basic and/or properties:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
—Václav Havel (b. 1936)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)