scholarly journals Möbius convolutions and the Riemann hypothesis

2005 ◽  
Vol 2005 (22) ◽  
pp. 3599-3608 ◽  
Author(s):  
Luis Báez-Duarte
Keyword(s):  
Real Axis ◽  
Riemann Hypothesis ◽  
Entire Functions ◽  
General Theorem ◽  
Positive Real Axis ◽  
Positive Real ◽  
Necessary And Sufficient Criteria ◽  
Necessary And Sufficient

The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and of Hardy and Littlewood, based on the order of certain entire functions on the positive real axis, are here embedded in a general theorem for a class of entire functions, which in turn is seen to be a consequence of a rather transparent convolution criterion. Some properties of the convolutions involved sharpen what is hitherto known for the Riesz function.

scholarly journals An Extension of the Almost Isolated Singularity of Finite Exponential Order

1964 ◽  
Vol 14 (2) ◽  
pp. 137-141
Author(s):  
R. Wilson
Keyword(s):  
Real Axis ◽  
Taylor Series ◽  
Positive Real Axis ◽  
Isolated Singularity ◽  
Positive Real ◽  
Small Disk ◽  
Exponential Order ◽  
Disk Centre ◽  
Image Position

Let f(z) be represented on its circle of convergence |z| = 1 by the Taylor seriesand suppose that its sole singularity on |z| = 1 is an almost isolated singularity at z = 1. In the neighbourhood of such a singularity f(z) is regular on a sufficiently small disk, centre z = 1, with the outward drawn radius along the positive real axis excised. If also in this neighbourhood |f(z)| e−(1/δ)ρ remains bounded for some finite ρ, where δ is the distance from the excised radius, then the singularity is said to be of finite exponential order.

scholarly journals Integral representations of Mittag-Leffler function on the positive real axis

2019 ◽  
Vol 20 (2) ◽  
pp. 217
Author(s):  
Eliana Contharteze Grigoletto ◽  
Edmundo Capelas Oliveira ◽  
Rubens Figueiredo Camargo
Keyword(s):  
Fractional Calculus ◽  
Laplace Transform ◽  
Real Axis ◽  
Integral Representations ◽  
Positive Real Axis ◽  
Inverse Laplace Transform ◽  
Positive Real

The Mittag-Leffler functions appear in many problems associated with fractional calculus. In this paper, we use the methodology for evaluation of the inverse Laplace transform, proposed by M. N. Berberan-Santos, to show that the three-parameter Mittag-Leffler function has similar integral representations on the positive real axis. Some of the integrals are also presented.

On the Radius of Curvature for Convex Analytic Functions

1970 ◽  
Vol 22 (3) ◽  
pp. 486-491 ◽  
Author(s):  
Paul Eenigenburg
Keyword(s):  
Real Axis ◽  
Jordan Curve ◽  
Convex Functions ◽  
Analytic Functions ◽  
Tangent Line ◽  
Positive Real Axis ◽  
Radius Of Curvature ◽  
Arc Length ◽  
Positive Real ◽  
Image Position

Definition 1.1. Let be analytic for |z| < 1. If ƒ is univalent, we say that ƒ belongs to the class S.Definition 1.2. Let ƒ ∈ S, 0 ≦ α < 1. Then ƒ belongs to the class of convex functions of order α, denoted by Kα, provided(1)and if > 0 is given, there exists Z0, |Z0| < 1, such thatLet ƒ ∈ Kα and consider the Jordan curve ϒτ = ƒ(|z| = r), 0 < r < 1. Let s(r, θ) measure the arc length along ϒτ; and let ϕ(r, θ) measure the angle (in the anti-clockwise sense) that the tangent line to ϒτ at ƒ(reiθ) makes with the positive real axis.

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