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.

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

scholarly journals Second variation of Selberg zeta functions and curvature asymptotics

2019 ◽  
Vol 57 (1) ◽  
pp. 23-60
Author(s):  
Ksenia Fedosova ◽  
Julie Rowlett ◽  
Genkai Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Vector Bundle ◽  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Second Variation ◽  
Selberg Zeta Function ◽  
The Second Variation

Abstract We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, Z(s), on Teichmüller space. We then use this formula to determine the asymptotic behavior as $$\mathfrak {R}s \rightarrow \infty $$Rs→∞ of the second variation. As a consequence, for $$m \in {\mathbb {N}}$$m∈N, we obtain the complete expansion in m of the curvature of the vector bundle $$H^0(X_t, {\mathcal {K}}_t)\rightarrow t\in {\mathcal {T}}$$H0(Xt,Kt)→t∈T of holomorphic m-differentials over the Teichmüller space $${\mathcal {T}}$$T, for m large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $$O(m^2 \mathrm{e}^{-l_0 m}),$$O(m2e-l0m), where $$l_0$$l0 is the length of the shortest closed hyperbolic geodesic.

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).

Inverse Problems for Partition Functions

2001 ◽  
Vol 53 (4) ◽  
pp. 866-896
Author(s):  
Yifan Yang
Keyword(s):  
Asymptotic Behavior ◽  
Partition Function ◽  
Inverse Problems ◽  
Large Class ◽  
Generating Function ◽  
Riemann Hypothesis ◽  
Arithmetic Function ◽  
Partition Functions ◽  
Class Of Functions ◽  
Weighted Partition

AbstractLet pw(n) be the weighted partition function defined by the generating function , where w(m) is a non-negative arithmetic function. Let be the summatory functions for pw(n) and w(n), respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions Φ(u) and λ(u), an estimate for Pw(u) of the formlog Pw(u) = Φ(u){1 + Ou(1/λ(u))} (u→∞) implies an estimate forNw(u) of the formNw(u) = Φ*(u){1+O(1/ log ƛ(u))} (u→∞) with a suitable function Φ*(u) defined in terms of Φ(u). We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

Generating Functions for a Class of Arithmetic Functions

1966 ◽  
Vol 9 (4) ◽  
pp. 427-431 ◽  
Author(s):  
A. A. Gioia ◽  
M.V. Subbarao
Keyword(s):  
Generating Functions ◽  
Multiplicative Function ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Prime Divisors ◽  
Image Position

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

scholarly journals Mean value theorems for a class of density-like arithmetic functions

2020 ◽  
pp. 1-15
Author(s):  
Lucas Reis
Keyword(s):  
Finite Field ◽  
Arithmetic Function ◽  
Field Extension ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Mean Value Theorem ◽  
Primitive Elements ◽  
Mean Values ◽  
Mean Value Theorems ◽  
Nonzero Mean

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.

Partial Euler Products on the Critical Line

2005 ◽  
Vol 57 (2) ◽  
pp. 267-297 ◽  
Author(s):  
Keith Conrad
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Function Fields ◽  
Number Fields ◽  
Euler Product ◽  
Second Moments ◽  
Precise Sense ◽  
Euler Products

AbstractThe initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seemsmuch deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

scholarly journals On asymptotic behavior of Dirichlet inverse

2020 ◽  
Vol 16 (06) ◽  
pp. 1337-1354
Author(s):  
Falko Baustian ◽  
Vladimir Bobkov
Keyword(s):  
Asymptotic Behavior ◽  
Arithmetic Function ◽  
Upper Bounds ◽  
Dirichlet Convolution

Let [Formula: see text] be an arithmetic function with [Formula: see text] and let [Formula: see text] be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behavior of [Formula: see text] with regard to the asymptotic behavior of [Formula: see text] assuming that the latter one grows or decays with at most polynomial or exponential speed. As a by-product, we obtain simple but constructive upper bounds for the number of ordered factorizations of [Formula: see text] into [Formula: see text] factors.

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