scholarly journals Zeta functions of analytic rings via Euler products

1986 ◽  
Vol 62 (5) ◽  
pp. 193-196
Author(s):  
Nobushige Kurokawa
Keyword(s):  
Zeta Functions ◽  
Euler Products

Shimura Varieties and the Selberg Trace Formula

1977 ◽  
Vol 29 (6) ◽  
pp. 1292-1299 ◽  
Author(s):  
R. P. Langlands
Keyword(s):  
Trace Formula ◽  
Automorphic Forms ◽  
Zeta Functions ◽  
Higher Dimensions ◽  
Shimura Varieties ◽  
Selberg Trace Formula ◽  
Work In Progress ◽  
Euler Products ◽  
Dimension One

This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an attempt to continue to higher dimensions the study of the relation between the Hasse-Weil zeta-functions of Shimura varieties and the Euler products associated to automorphic forms, which was initiated by Eichler, and extensively developed by Shimura for the varieties of dimension one bearing his name. The method used has its origins in an idea of Sato, which was exploited by Ihara for the Shimura varieties associated to GL(2).

RUELLE TYPE ZETA FUNCTIONS FOR TORI AND SOME ARITHMETICS

2004 ◽  
Vol 15 (07) ◽  
pp. 691-715 ◽  
Author(s):  
NOBUSHIGE KUROKAWA ◽  
MASATO WAKAYAMA
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Distribution ◽  
Arithmetic Function ◽  
Zeta Functions ◽  
Arithmetic Functions ◽  
Length Functions ◽  
Euler Products

We introduce various Ruelle type zeta functions ζL(s) according to a choice of homogeneous "length functions" for a lattice L in [Formula: see text] via Euler products. The logarithm of each ζL(s) yields naturally a certain arithmetic function. We study the asymptotic distribution of averages of such arithmetic functions. Asymptotic behavior of the zeta functions at the origin s=0 are also investigated.

scholarly journals Natural boundaries for Euler products of Igusa zeta functions of elliptic curves

2018 ◽  
Vol 14 (08) ◽  
pp. 2317-2331
Author(s):  
Marcus du Sautoy
Keyword(s):  
Elliptic Curves ◽  
Zeta Functions ◽  
Symmetric Power ◽  
Meromorphic Continuation ◽  
Natural Boundary ◽  
Integer Polynomials ◽  
Analytic Behavior ◽  
Euler Products ◽  
Natural Boundaries

We study the analytic behavior of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato–Tate conjectures, we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.

scholarly journals ZETA FUNCTIONS OF GROUPS: EULER PRODUCTS AND SOLUBLE GROUPS

2002 ◽  
Vol 45 (1) ◽  
pp. 149-154
Author(s):  
Marcus du Sautoy
Keyword(s):  
Zeta Functions ◽  
Nilpotent Groups ◽  
Subject Classification ◽  
Euler Product ◽  
Soluble Groups ◽  
Euler Products ◽  
Primary 20F16 ◽  
Zeta Functions Of Groups

AbstractThe well-behaved Sylow theory for soluble groups is exploited to prove an Euler product for zeta functions counting certain subgroups in pro-soluble groups. This generalizes a result of Grunewald, Segal and Smith for nilpotent groups.AMS 2000 Mathematics subject classification: Primary 20F16; 11M99

scholarly journals Convergence of the zeta functions of prehomogeneous vector spaces

2003 ◽  
Vol 170 ◽  
pp. 1-31 ◽  
Author(s):  
Hiroshi Saito
Keyword(s):  
Zeta Functions ◽  
Algebraic Number Field ◽  
Connected Components ◽  
Singular Set ◽  
Connected Component ◽  
Generic Point ◽  
Prehomogeneous Vector Space ◽  
Prehomogeneous Vector Spaces ◽  
Euler Products ◽  
Local Zeta Functions

AbstractLet (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

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