G. Mora and Y. Cherruault
This paper seeks to present an original method for transforming multiple integrals into simple integrals.
Abstract
Purpose
This paper seeks to present an original method for transforming multiple integrals into simple integrals.
Design/methodology/approach
This can be done by using α‐dense curves invented by Y. Cherruault and A. Guillez at the beginning of the 1980s.
Findings
These curves allow one to approximate the space Rn (or a compact of Rn) with the accuracy α. They generalize fractal curves of Mandelbrobdt. They can be applied to global optimization where the multivariables functional is transformed into a functional depending on a single variable.
Practical implications
Applied to a multiple integral, the α‐dense curves using Chebyshev's kernels permit one to obtain a simple integral approximating the multiple integral. The accuracy depends on the choice of α.
Originality/value
The paper presents an original method for transforming integrals into simple integrals.
Details
Keywords
G. Mora, G. Mora‐Porta and J.M. Sepulcre
This paper aims to introduce a new class of entire functions whose zeros (zk)k≥1 satisfy ∑k=1∞Im zk=O(1).
Abstract
Purpose
This paper aims to introduce a new class of entire functions whose zeros (zk)k≥1 satisfy ∑k=1∞Im zk=O(1).
Design/methodology/approach
This is done by means of a Ritt's formula which is used to prove that every partial sum of the Riemann Zeta function, ζn(z):=∑k=1n1/kz, n≥2, has zeros (snk)k≥1 verifying ∑k=1∞Re snk=O(1) and extending this property to a large class of entire functions denoted by AO.
Findings
It is found that this new class AO has a part in common with the class A introduced by Levin but is distinct from it. It is shown that, in particular, AO contains every partial sum of the Riemann Zeta function ζn(iz) and every finite truncation of the alternating Dirichlet series expansion of the Riemann zeta function, Tn(iz):=∑k=1n(−1)k−1/kiz, for all n≥2.
Practical implications
With the exception of the n=2 case, numerical experiences show that all zeros of ζn(z) and Tn(z) are not symmetrically distributed around the imaginary axis. However, the fact consisting of every function ζn(iz) and Tn(iz) to be in the class AO implies the existence of a very precise physical equilibrium between the zeros situated on the left half‐plane and the zeros situated on the right half‐plane of each function. This is a relevant fact and it points out that there is certain internal rule that distributes the zeros of ζn(z) and Tn(z) in such a way that few zeros on the left of the imaginary axis and far away from it, must be compensated with a lot of zeros on the right of the imaginary axis and close to it, and vice versa.
Originality/value
The paper presents an original class of entire functions that provides a new point of view to study the approximants and the alternating Dirichlet truncations of the Riemann zeta function.