transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
The Dedekind zeta function is a generalization of the Riemann zeta function from the rational numbers/integers to number fields/their rings of integers.
The analog for function fields is the Weil zeta function, while the generalization to higher dimensional arithmetic geometry are the arithmetic zeta functions.
For a number field then all special values of the Dedekind zeta function for integer happen to be periods (MO comment).
Based on this (Deligne 79) identified critical values of -functions (at certain integers) and conjectured that these are all these are algebraic multiples of determinants of matrices whose entries are periods. For more on this see at special values of L-functions.
The Dedekind zeta function has an adelic integral expression in direct analogy to that of the Riemann zeta function. This is due to (Tate 50), highlighted by Ivan Fesenko in (Goldfeld-Hundley 11, Interlude remark (1)).
The functional equation of the Dedekind zeta function follows from its adelic integral representation in direct analogy to how this works for the Riemann zeta function. This is due to (Tate 50), highlighted by Ivan Fesenko in (Goldfeld-Hundley 11, Interlude remark (1)).
The Dedekind zeta function of has a simple pole at . The class number formula says that its residue there is proportional the the product of the regulator of with the class number of
The Dedekind zeta function is the Artin L-function for trivial Galois representation
See at Artin L-function – Relation with Dedekind zeta function.
The Dedekind zeta function has an expression as an integral over a kernel given by a Hecke theta function (Kowalski (2.3) (2.4)).
The Dedekind zeta function of is equivalently the Hasse-Weil zeta function of .
The function field analogy in view of the discussion at zeta function of an elliptic differential operator says that the Dedekind zeta function is analogous to the regulated functional trace of a would-be “Dirac operator on Spec(Z)”.
Other identifications/analogies of the Riemann zeta function (and more generally the Dedekind zeta-function) with partition functions in physics have been proposed, in particular the Bost-Connes system.
John Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–34 1950
Wikipedia, Dedekind zeta function
E. Kowalski, section 1.4 of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)
Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)
Last revised on September 1, 2014 at 09:10:59. See the history of this page for a list of all contributions to it.