scholarly journals A Riemann–von Mangoldt-Type Formula for the Distribution of Beurling Primes

2021 ◽  
Author(s):  
Szilárd Gy. Révész
Keyword(s):  
Zeta Function ◽  
Integral Transformation ◽  
Logarithmic Derivative ◽  
Distribution Of Zeros ◽  
Line Contour ◽  
Axiom A ◽  
Type Formula ◽  
Von Mangoldt Function ◽  
Generalized System ◽  
Arithmetical Semigroup

In this paper we work out a Riemann–von Mangoldt type formula for the summatory function := , where is an arithmetical semigroup (a Beurling generalized system of integers) and is the corresponding von Mangoldt function attaining with a prime element and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function , belonging to , to the number of zeroes of in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of , under the sole additional assumption that Knopfmacher’s Axiom A is satisfied. We also construct a technically useful broken line contour to which the technic of integral transformation can be well applied. The whole work serves as a first step towards a further study of the distribution of zeros of the Beurling zeta function, providing appropriate zero density and zero clustering estimates, to be presented in the continuation of this paper.

Analysis on arithmetic schemes. II

2010 ◽  
Vol 5 (3) ◽  
pp. 437-557 ◽  
Author(s):  
Ivan Fesenko
Keyword(s):  
Zeta Function ◽  
Logarithmic Derivative ◽  
Boundary Function ◽  
Boundary Integral ◽  
Integration Theory ◽  
Meromorphic Continuation ◽  
Two Dimensional ◽  
Fundamental Properties ◽  
Arithmetic Surfaces ◽  
Dimensional Version

AbstractWe construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K2-delic and K1×K1-delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.

scholarly journals On the distribution of zeros of a Ruelle zeta-function

1994 ◽  
Vol 159 (3) ◽  
pp. 433-441 ◽  
Author(s):  
A. Eremenko ◽  
G. Levin ◽  
M. Sodin
Keyword(s):  
Zeta Function ◽  
Distribution Of Zeros ◽  
Ruelle Zeta Function

REGULARIZATION OF GENERAL MULTIDIMENSIONAL EPSTEIN ZETA-FUNCTIONS

1989 ◽  
Vol 01 (01) ◽  
pp. 113-128 ◽  
Author(s):  
E. ELIZALDE ◽  
A. ROMEO
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Partition Function ◽  
Zeta Function ◽  
Finite Temperature ◽  
Casimir Effect ◽  
Zeta Functions ◽  
Quantum Field ◽  
Type Formula

We study expressions for the regularization of general multidimensional Epstein zeta-functions of the type [Formula: see text] After reviewing some classical results in the light of the extended proof of zeta-function regularization recently obtained by the authors, approximate but very quickly convergent expressions for these functions are derived. This type of analysis has many interesting applications, e.g. in any quantum field theory defined in a partially compactified Euclidean spacetime or at finite temperature. As an example, we obtain the partition function for the Casimir effect at finite temperature.

scholarly journals Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables

2019 ◽  
Vol 16 (2) ◽  
pp. 154-180
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Holomorphic Function ◽  
Logarithmic Derivative ◽  
Holomorphic Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Necessary And Sufficient Condition ◽  
Positive Continuous Function ◽  
Distribution Of Zeros ◽  
Necessary And Sufficient ◽  
Class Of Functions

We investigate the slice holomorphic functions of several complex variables that have a bounded \(L\)-index in some direction and are entire on every slice \(\{z^0+t\mathbf{b}: t\in\mathbb{C}\}\) for every \(z^0\in\mathbb{C}^n\) and for a given direction \(\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}\). For this class of functions, we prove some criteria of boundedness of the \(L\)-index in direction describing a local behavior of the maximum and minimum moduli of a slice holomorphic function and give estimates of the logarithmic derivative and the distribution of zeros. Moreover, we obtain analogs of the known Hayman theorem and logarithmic criteria. They are applicable to the analytic theory of differential equations. We also study the value distribution and prove the existence theorem for those functions. It is shown that the bounded multiplicity of zeros for a slice holomorphic function \(F:\mathbb{C}^n\to\mathbb{C}\) is the necessary and sufficient condition for the existence of a positive continuous function \(L: \mathbb{C}^n\to\mathbb{R}_+\) such that \(F\) has a bounded \(L\)-index in direction.

scholarly journals Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems

1996 ◽  
Vol 16 (4) ◽  
pp. 805-819 ◽  
Author(s):  
Hans Henrik Rugh
Keyword(s):  
Dynamical Systems ◽  
Zeta Function ◽  
Complex Plane ◽  
Entire Functions ◽  
Fredholm Determinant ◽  
Three Dimensional ◽  
Zeta Functions ◽  
Selberg Zeta Function ◽  
Fredholm Determinants ◽  
Axiom A

AbstractWe consider a generalized Fredholm determinant d(z) and a generalized Selberg zeta function ζ(ω)−1 for Axiom A diffeomorphisms of a surface and Axiom A flows on three-dimensional manifolds, respectively. We show that d(z) and ζ(ω)−1 extend to entire functions in the complex plane. That the functions are entire and not only meromorphic is proved by a new method, identifying removable singularities by a change of Markov partitions.

New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants

2008 ◽  
Vol 464 (2091) ◽  
pp. 711-731 ◽  
Author(s):  
Mark W Coffey
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Logarithmic Derivative ◽  
Bernoulli Numbers ◽  
Stieltjes Constants ◽  
The Riemann Zeta Function ◽  
Power Series Expansions ◽  
Riemann Zeta ◽  
Derivatives Of ◽  
Logarithmic Derivatives

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. A new representation of λ k is developed in terms of the Stieltjes constants γ j and the subcomponent sums are discussed and analysed. Accompanying this decomposition, we find a new representation of the constants η j entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about s =1. We also demonstrate that the η j coefficients are expressible in terms of the Bernoulli numbers and certain other constants. We determine new properties of η j and σ j , where are the sums of reciprocal powers of the non-trivial zeros of the Riemann zeta function.

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