scholarly journals On certain sums over ordinates of zeta-zeros II

2019 ◽  
Author(s):  
Andriy Bondarenko ◽  
Aleksandar Ivić ◽  
Eero Saksman ◽  
Kristian Seip
Keyword(s):  
Analytic Continuation ◽  
Critical Line ◽  
Laurent Expansion ◽  
Second Moment ◽  
International Audience ◽  
Complex Zeros ◽  
Gamma S

International audience Let γ denote the imaginary parts of complex zeros ρ = β + iγ of ζ(s). The problem of analytic continuation of the function $G(s) :=\sum_{\gamma >0} {\gamma}^{-s}$ to the left of the line $\Re{s} = −1 $ is investigated, and its Laurent expansion at the pole s = 1 is obtained. Estimates for the second moment on the critical line $\int_{1}^{T} {| G (\frac{1}{2} + it) |}^2 dt $ are revisited. This paper is a continuation of work begun by the second author in [Iv01].

scholarly journals Analytic continuation of the critical line: Suggestions for QCD

2009 ◽  
Vol 80 (3) ◽  
Author(s):  
Paolo Cea ◽  
Leonardo Cosmai ◽  
Massimo D’Elia ◽  
Chiara Manneschi ◽  
Alessandro Papa
Keyword(s):  
Analytic Continuation ◽  
Critical Line

scholarly journals THE SCALAR BOX INTEGRAL AND THE MELLIN–BARNES REPRESENTATION

2011 ◽  
Vol 26 (15) ◽  
pp. 2557-2568 ◽  
Author(s):  
P. VALTANCOLI
Keyword(s):  
Analytic Continuation ◽  
Hypergeometric Functions ◽  
Laurent Expansion

We solve exactly the scalar box integral using the Mellin–Barnes representation. First we recognize the hypergeometric functions resumming the series coming from the scalar integrals, then we perform an analytic continuation before applying the Laurent expansion in ϵ = (d-4)/2 of the result.

scholarly journals The a-points of Faber polynomials for a special function

1970 ◽  
Vol 11 (1) ◽  
pp. 1-6
Author(s):  
Hassoon S. Al-Amiri
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Analytic Continuation ◽  
Special Function ◽  
Laurent Expansion ◽  
Faber Polynomials ◽  
Analytic Element ◽  
Formal Expansion ◽  
Image Position

Let f(ζ) be a power series of the formwhere lim sup |an|1/n < ∞. The Faber polynomials {fn(ζ)} (n = 0, 1, 2, …) are the polynomial parts of the formal expansion of (f(ζ))n about ζ = ∞. Series (1) defines an analytic element of an analytic function which we designate as w = f(ζ). Since at ζ = ∞ the analytic element is univalent in some neighborhood of infinity; thus the inverse of this element is uniquely determined in some neighborhood of w= ∞, and has a Laurent expansion of the formwhere lim sup |bn|1/n = p < ∞. Let ζ = g(w) be this single-valued function defined by (2) in |w| > p. No analytic continuation of g(w) will be considered.

scholarly journals Analytic continuation in two-color QCD: new results on the critical line

2009 ◽  
Vol 820 (1-4) ◽  
pp. 239c-242c ◽  
Author(s):  
P. Cea ◽  
L. Cosmai ◽  
M. D'Elia ◽  
A. Papa
Keyword(s):  
Analytic Continuation ◽  
Critical Line

scholarly journals On complex zeros off the critical line for non-monomial polynomial of zeta-functions

2016 ◽  
Vol 284 (1-2) ◽  
pp. 23-39 ◽  
Author(s):  
Takashi Nakamura ◽  
Łukasz Pańkowski
Keyword(s):  
Critical Line ◽  
Zeta Functions ◽  
Complex Zeros

scholarly journals An estimate for the Mellin transform of powers of Hardy's function.

2010 ◽  
Vol Volume 33 ◽  
Author(s):  
Matti Jutila
Keyword(s):  
Entire Function ◽  
Zeta Function ◽  
Mellin Transform ◽  
Critical Line ◽  
Classical Estimate ◽  
International Audience ◽  
Hardy’S Function

International audience We show that a certain modified Mellin transform $\mathcal M(s)$ of Hardy's function is an entire function. There are reasons to connect $\mathcal M(s)$ with the function $\zeta(2s-1/2)$, and then the orders of $\mathcal M(s)$ and $\zeta(s)$ should be comparable on the critical line. Indeed, an estimate for $\mathcal M(s)$ is proved which in the particular case of the critical line coincides with the classical estimate of the zeta-function.

scholarly journals Newton Polytopes of Cluster Variables of Type $A_n$

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Adam Kalman
Keyword(s):  
Type A ◽  
Laurent Expansion ◽  
Cluster Algebras ◽  
Newton Polytope ◽  
Newton Polytopes ◽  
Affine Hull ◽  
Cluster Variable ◽  
Nous Montrons ◽  
The Face ◽  
International Audience

International audience We study Newton polytopes of cluster variables in type $A_n$ cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type $A$ cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph. Nous étudions polytopes de Newton des variables amassées dans les algèbres amassées de type A, dont les variables sont indexés par les diagonales et les côtés d’un polygone. Nos principaux résultats comprennent une description explicite de l’enveloppe affine et facettes du polytope de Newton du développement de Laurent de toutes variables amassées. En particulier, nous montrons que tout monôme Laurent dans un développement de Laurent de variable amassée de type A correspond à un sommet du polytope de Newton. Nous décrivons aussi le treillis des facesde chaque polytope de Newton via un isomorphisme avec le treillis des sous-graphes élémentaires du “snake graph” qui est associé.

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