The Exponential Function Written as Split Infinite Product | Chapter 10 | Research and Applications Towards Mathematics and Computer Science Vol. 5

 Split limitless polynomial products have per description all their roots on the evident and imaginary axes. The invented axe fluctuated to the critical line, all the ancestries on this axe are switched as well. This is genuine for the gamma and the zeta functions expressed as limitless polynomial products also. This opens the possibility to confirm the placement of the roots of the zeta function from Riemann on the fault-finding line. The product of four limitless polynomials—two with all fictitious roots and two accompanying all real ancestries—can be created by dividing any polynomial presented as an infinite output with all helpful real ancestries into two equal halves. Such infinite fruit define adjoint limitless polynomials with ancestries on adjacent (actual and imaginary) ancestries by equations. It is displayed that changing the relates of one of the abutting axes to a parallel line does not affect the ancestries' relative arrangement because they are transferred to the parallel line. There are evaluated the general relationships 'tween the original and adjoint polynomials. These relations are statement representations of the connections of Euler and Pythagoras in form of limitless polynomial products. They are hereditary properties of split polynomial production. If the shifting of the standards for judging or deciding corresponds to the fluctuating of the imaginary axes to the detracting line, then the connections of Euler take the form corresponding to their incident in the functional equating of the Riemann zeta function: the roots on the fictitious axes are all shifted to the detracting line. Since it is known that the gamma and the zeta functions grant permission be written as collected functions with epidemic and trigonometric parts, this opens the likelihood to prove the installation of the zeta function on the critical line.

Author(s) Details:

Pál Doroszlai,
Fö utca 25, 8254 Kékkut, Hungary.

Horacio Keller,
Swiss Federal Institute of Technology, Switzerland.

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