Transformation formulas associated with integrals involving the Riemann Ξ-function

Using residue calculus and the theory of Mellin transforms, we evaluate integrals of a certain type involving the Riemann Ξ-function, which give transformation formulas of the form F(z, α) = F(z, β), where αβ = 1. This gives a unified approach for generating certain modular transformation formulas, including a famous formula of Ramanujan and Guinand.     

Series transformations and integrals involving the Riemann Ξ-function

The transformation formulas of Ramanujan, Hardy, Koshliakov and Ferrar are unified, in the sense that all these formulas come from the same source, namely, a general formula involving an integral of Riemann's Ξ-function. We give proofs of all of these transformation formulas using the theory of Mellin transforms and the residue theorem. Our study includes new extensions of the formulas of Koshliakov and Ferrar through their connection with integrals involving the Riemann Ξ-function.     

On the zeros of approximations of the Ramanujan Ξ-function

The analogue of the Riemann hypothesis for the Ramanujan zeta function states that all zeros of the Ramanujan Ξ-function have real zeros only. We study the zeros of approximations of the Ramanujan Ξ-function. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00021).     

Quadrature formulas with exponential convergence and calculation of the Fermi–Dirac integrals

A class of functions for which the trapezoidal rule has superpower convergence is described: these are infinitely differentiable functions all of whose odd derivatives take equal values at the left and right endpoints of the integration interval. An heuristic law is revealed; namely, the convergence exponentially depends on the number of nodes, and the exponent equals the ratio of the length of integration interval to the distance from this interval to the nearest pole of the integrand. On the basis of the obtained formulas, a method for calculating the Fermi–Dirac integrals of half-integer order is proposed, which is substantially more economical than all known computational methods. As a byproduct, an asymptotics of the Bernoulli numbers is found.     

Analogues of Feynman formulas for ill-posed problems associated with the Schrödinger equation

Representations of Schrödinger semigroups and groups by Feynman iterations are studied. The compactness, rather than convergence, of the sequence of Feynman iterations is considered. Approximations of solutions of the Cauchy problem for the Schrödinger equation by Feynman iterations are investigated. The Cauchy problem for the Schrödinger equation under consideration is ill-posed. From the point of view of the approach of the paper, this means that the problem has no solution in the sense of integral identity for some initial data. The well-posedness of the Cauchy problem can be recovered by extending the operator to a selfadjoint one; however, there exists continuum many such extensions. Feynman iterations whose partial limits are the solutions of all Cauchy problems obtained for various self-adjoint extensions are studied.     

On the convergence of multipoint Padé-type approximants and quadrature formulas associated with the unit circle

We study the convergence of rational interpolants with prescribed poles on the unit circle to the Herglotz-Riesz transform of a complex measure supported on [–, ]. As a consequence, quadrature formulas arise which integrate exactly certain rational functions. Estimates of the rate of convergence of these quadrature formulas are also included.This research was performed as part of the European project ROLLS under contract CHRX-CT93-0416.     

Riemann–Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue Spaces

The present paper deals with a solution of the Riemann–Hilbert problem with piecewise continuous coefficients in the class of Cauchy type integrals with densities in grand Lebesgue spaces. Necessary and sufficient solvability condition is established. In solvability case the solutions in explicit form are constructed.     

Representation of the β-function and anomalous dimensions by nonsingular integrals: Proof of the main relation

A method for calculating the β-function and anomalous dimensions, convenient for numerical calculations in the ?-expansion framework, was previously proposed, and the relation underlying the method was verified up to the four-loop approximation. We prove this relation in all orders of the perturbation theory.     

The Schatten–von Neumann class associated with the Gabor–Riemann–Liouville operator

Periodica Mathematica Hungarica - In this paper, we define the localization operator associated with the Riemann–Liouville operator, and show that it is not only bounded, but it is also in...     

Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions

Riemann conjectured that all the zeros of the Riemann ≡-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums ≡ _N (z) in Riemann’s uniformly convergent infinite series expansion of ≡(z) involving incomplete gamma functions. We conjecture that when the zeros of ≡ _N (z) in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this conjecture implies the RH, and discuss some computational evidence for this and other related conjectures.     

On the Riemann–Liouville Operators with Variable Limits

Siberian Mathematical Journal -     

Solvability of fully nonlinear functional equations involving Erdélyi-Kober fractional integrals on the unbounded interval

In this paper, we study the existence of solutions to fully nonlinear functional equations involving Erdélyi-Kober fractional integrals on the unbounded interval. By constructing a closed and convex subset of a Fréchet space, sufficient conditions are presented by using a fixed point theorem via the Kuratowski measure. Finally, an example is given to illustrate the obtained result.     

Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation

We study the one-dimensional periodic derivative nonlinear Schrödinger equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion \({\textstyle \int }h_k\), \(k\in {\mathbb {Z}}_{+}\). In each \({\textstyle \int }h_{2k}\) the term with the highest regularity involves the Sobolev norm \(\dot{H}^{k}({\mathbb {T}})\) of the solution of the DNLS equation. We show that a functional measure on \(L^2({\mathbb {T}})\), absolutely continuous w.r.t. the Gaussian measure with covariance \(({\mathbb {I}}+(-\varDelta )^{k})^{-1}\), is associated to each integral of motion \({\textstyle \int }h_{2k}\), \(k\ge 1\).     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

Some bounds for integrals with refinements of the Grüss inequality

Some integral bounds are obtained which provide refinements and extensions of the Grüss type inequalities. We also get an alternative formulation of the Grüss inequality.