On the Zeros of Euler Product Dirichlet Functions

Ghişa, Dorin

doi:10.4236/apm.2019.912048

Cited by 3 publications

(1 citation statement)

References 2 publications

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“…From it, Euler polynomials ( ) We think that this work has some significative value for the comprehension of the Euler polynomials. Nevertheless it may be completed with many works on them, one may find in literature for example by Vergara-Hermosilla [16] concerning the properties of Hurwitz polynomials and by Ghisa, D. [17] concerning Euler Product Dirichlet Functions.…”

Section: Remarks and Conclusionmentioning

confidence: 99%

Obtaining Simply Explicit Form and New Properties of Euler Polynomials by Differential Calculus

Si¹

2023

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Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums; may be all its relations with Bernoulli polynomials, Bernoulli numbers; its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained; the formulae for obtaining all m π as series on m k − and for expanding functions into series of Euler polynomials.

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Section: Remarks and Conclusionmentioning

confidence: 99%

Obtaining Simply Explicit Form and New Properties of Euler Polynomials by Differential Calculus

Si¹

2023

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The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

Ghişa¹

2020

APM

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The Cauchy integral formula expresses the value of a function () f z , which is analytic in a simply connected domain D, at any point 0 z interior to a simple closed contour C situated in D in terms of the values of () 0 f z z z − on C. We deal in this paper with the question whether C can be the boundary ∂Ω of a fundamental domain Ω of () f z. At the first look the answer appears to be negative since ∂Ω contains singular points of the function and it can be unbounded. However, the extension of Cauchy integral formula to some of these unbounded curves, respectively arcs ending in singular points of () f z is possible due to the fact that they can be obtained at the limit as r → ∞ of some bounded curves contained in the pre-image of the circle z r = and of some circles 1 z a r − = for which the formula is valid.

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On the Non-Trivial Zeros of Dirichlet Functions

Ghişa¹

2021

APM

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The purpose of this research is to extend to the functions obtained by meromorphic continuation of general Dirichlet series some properties of the functions in the Selberg class, which are all generated by ordinary Dirichlet series. We wanted to put to work the powerful tool of the geometry of conformal mappings of these functions, which we built in previous research, in order to study the location of their non-trivial zeros. A new approach of the concept of multiplier in Riemann type of functional equation was necessary and we have shown that with this approach the non-trivial zeros of the Dirichlet function satisfying a Reimann type of functional equation are either on the critical line, or they are two by two symmetric with respect to the critical line. The Euler product general Dirichlet series are defined, a wide class of such series is presented, and finally by using geometric and analytic arguments it is proved that for Euler product functions the symmetric zeros with respect to the critical line must coincide.

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On the Zeros of Euler Product Dirichlet Functions

Cited by 3 publications

References 2 publications

Obtaining Simply Explicit Form and New Properties of Euler Polynomials by Differential Calculus

Obtaining Simply Explicit Form and New Properties of Euler Polynomials by Differential Calculus

The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

On the Non-Trivial Zeros of Dirichlet Functions

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