scholarly journals Local zeta functions and Newton polyhedra

2003 ◽  
Vol 172 ◽  
pp. 31-58 ◽  
Author(s):  
W. A. Zuniga-Galindo
Keyword(s):  
Zeta Function ◽  
Valuation Ring ◽  
Zeta Functions ◽  
Newton Polyhedron ◽  
Group Of Units ◽  
Local Zeta Function ◽  
Degenerate Polynomial ◽  
Small Set ◽  
Local Zeta Functions ◽  
Almost All

AbstractTo a polynomial f over a non-archimedean local field K and a character χ of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z (s, f, χ). In this paper, we study the local zeta function Z(s, f, χ) associated to a non-degenerate polynomial f, by using an approach based on the p-adic stationary phase formula and Néron p-desingularization. We give a small set of candidates for the poles of Z (s, f, χ) in terms of the Newton polyhedron Γ(f) of f. We also show that for almost all χ, the local zeta function Z(s, f, χ) is a polynomial in q−s whose degree is bounded by a constant independent of χ. Our second result is a description of the largest pole of Z(s, f, χtriv) in terms of Γ(f) when the distance between Γ(f) and the origin is at most one.

Stability results for local zeta functions of groups algebras, and modules

2017 ◽  
Vol 165 (3) ◽  
pp. 435-444 ◽  
Author(s):  
TOBIAS ROSSMANN
Keyword(s):  
Group Theory ◽  
Functional Equations ◽  
Zeta Functions ◽  
Local Zeta Functions ◽  
Single Formula ◽  
Almost All ◽  
Stability Results ◽  
Zeta Functions Of Groups ◽  
Local Base

AbstractVarious types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

scholarly journals A Quadruple Integral Involving Chebyshev Polynomials Tn(x): Derivation and Evaluation

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 100
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Zeta Functions ◽  
Fundamental Constants ◽  
Zero Distribution ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Almost All

The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.

scholarly journals A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 9
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Parabolic Cylinder ◽  
Fundamental Constants ◽  
Lerch Zeta Function ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral ◽  
Almost All

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

scholarly journals A Note on the Summation of the Incomplete Gamma Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2369
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Zeta Functions ◽  
Incomplete Gamma Function ◽  
Zero Distribution ◽  
Special Values ◽  
Lerch Zeta Function ◽  
Almost All ◽  
Infinite Sum

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero distribution.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

scholarly journals A new generating function for calculating the Igusa local zeta function

2017 ◽  
Vol 304 ◽  
pp. 355-420 ◽  
Author(s):  
Raemeon A. Cowan ◽  
Daniel J. Katz ◽  
Lauren M. White
Keyword(s):  
Zeta Function ◽  
Generating Function ◽  
Local Zeta Function

scholarly journals MOTIVIC HILBERT ZETA FUNCTIONS OF CURVES ARE RATIONAL

2018 ◽  
Vol 19 (3) ◽  
pp. 947-964
Author(s):  
Dori Bejleri ◽  
Dhruv Ranganathan ◽  
Ravi Vakil
Keyword(s):  
Zeta Function ◽  
Generating Function ◽  
Zeta Functions ◽  
Hilbert Schemes ◽  
Grothendieck Ring

The motivic Hilbert zeta function of a variety $X$ is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points on $X$. In this paper, the motivic Hilbert zeta function of a reduced curve is shown to be rational.

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