scholarly journals On zeta functions associated to symmetric matrices III – An explicit form of L-Functions

1997 ◽  
Vol 146 ◽  
pp. 149-183 ◽  
Author(s):  
Tomoyoshi Ibukiyama ◽  
Hiroshi Saito
Keyword(s):  
Explicit Form ◽  
Zeta Functions ◽  
Symmetric Matrices ◽  
Special Values

Abstract.In [I-S2], we gave an explicit form of zeta functions associated to the space of symmetric matrices. In this paper, the case of L-functions is treated. In the case of definite symmetric matrices, we show the ratinality of special values of these L-functions.

scholarly journals On zeta functions associated to symmetric matrices, II: Functional equations and special values

2012 ◽  
Vol 208 ◽  
pp. 265-316 ◽  
Author(s):  
Tomoyoshi Ibukiyama ◽  
Hiroshi Saito
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Cusp Forms ◽  
Alternative Proof ◽  
Symmetric Matrices ◽  
Congruence Subgroups ◽  
Special Values ◽  
Dimension Formula ◽  
Integral Weight ◽  
The Matrix

AbstractNew simple functional equations of zeta functions of the prehomogeneous vector spaces consisting of symmetric matrices are obtained, using explicit forms of zeta functions in the previous paper, Part I, and real analytic Eisenstein series of half-integral weight. When the matrix size is 2, our functional equations are identical with the ones by Shintani, but we give here an alternative proof. The special values of the zeta functions at nonpositive integers and the residues are also explicitly obtained. These special values, written by products of Bernoulli numbers, are used to give the contribution of “central” unipotent elements in the dimension formula of Siegel cusp forms of any degree. These results lead us to a conjecture on explicit values of dimensions of Siegel cusp forms of any torsion-free principal congruence subgroups of the symplectic groups of general degree.

scholarly journals Fine Spectra of Tridiagonal Symmetric Matrices

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammed Altun
Keyword(s):  
Explicit Form ◽  
Resolvent Operator ◽  
Sequence Spaces ◽  
Symmetric Matrices ◽  
Infinite Matrices ◽  
Banded Matrices ◽  
Lower Triangular ◽  
The Resolvent Operator

The fine spectra of upper and lower triangular banded matrices were examined by several authors. Here we determine the fine spectra of tridiagonal symmetric infinite matrices and also give the explicit form of the resolvent operator for the sequence spaces , , , and .

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