Related papers: Change of variable in Riemann integration revisite…
In this note, we give a simple proof that the Riemann Hypothesis is unprovable in any reasonable axiom system.
A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby…
We mainly consider the general Caffarelli-Kohn-Nirenberg inequality in the Euclidean and Riemannian setting. In both cases, our proof relies mostly on a new parameter s conveniently introduced, see (2.7).
We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained…
A new parametric integral is obtained as a consequence of the Riemann hypothesis. An asymptotic multiplicability is the main property of this integral.
Laurin\v{c}ikas proposed that whether the universality theorem for the Riemann zeta-function shifted by an exponential function holds or not. This problem is solved for more general shifts by Andersson et al. In this paper, we generalize…
An analog of the Riemann hypothesis is proved in this paper. Some new integral equations for the functions $\pi(x)$ and $R(x)$ follows. A new effect that is shown is that these function - with essentially different behavior - are the…
We present a modification of Riesz's construction of the Lebesgue integral, leading directly to finite or infinite integrals, at the same time simplifying the proofs.
This note provides a new approach to a result of Foregger and related earlier results by Keilson and Eberlein. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the…
A multidimensional generalization of the Bernstein class of functions and the properties of functions of the introduced class are examined. In particular, a new proof of the integral representation of Bernstein functions of many variables…
In comparison with the previous version of this paper, the Introduction is slightly changed and some minor typos are deleted. All results are unchanged.
The authors modify the Balazard-Saias criterion for the Riemann hypothesis by changing the cutoff of the Dirichlet series. They establish some asymptotic results related to the modified criterion.
We are pleased to see that Jeckelmann has made many changes to the original version of his comment on our paper as a result of our response. Here is a copy of this powerful response that reveals problems in his previous results. However,…
We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure…
We present a proof of Hadamard Inverse Function Theorem by the methods of Variational Analysis, adapting an idea of I. Ekeland and E. Sere.
This note presents a new equivalence to the Riemann Hypothesis by means of the Salem integral equation.
This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and…
We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized…
In this paper, new inequalities connected with the celebrated Steffensen's integral inequality are proved.
We develop further the theory of integrable functions within the theory of relative simplicial motivic measures. We provide a primitive change of variables formula for this theory.