The Exponential Function Written as Split Infinite Product - Doroszlai , Pál; Keller , Horacio (Research and Applications Towards Mathematics and Computer Science Vol. 5, 2023)
Split infinite polynomial products have per definition all their roots on the real and imaginary axes. The imaginary axe shifted to the critical line, all the roots on this axe are shifted as well. This is valid for the gamma and the zeta functions expressed as infinite polynomial products as well. This opens the possibility to prove the placement of the roots of the zeta function from Riemann on the critical line. The product of four infinite polynomials—two with all imaginary roots and two with all real roots—can be created by splitting any polynomial represented as an infinite product with all positive real roots into two equal halves. Such infinite products define adjoint infinite polynomials with roots on adjacent (real and imaginary) roots by equations. It is demonstrated that changing the coordinates of one of the adjacent axes to a parallel line does not affect the roots' relative arrangement since they are transferred to the parallel line. There are assessed the general relationships between the original and adjoint polynomials. These relations are generalized representations of the relations of Euler and Pythagoras in form of infinite polynomial products. They are inherent properties of split polynomial products. If the shifting of the coordinate system corresponds to the shifting of the imaginary axes to the critical line, then the relations of Euler take the form corresponding to their occurrence in the functional equation of the Riemann zeta function: the roots on the imaginary axes are all shifted to the critical line. Since it is known that the gamma and the zeta functions may be written as composed functions with exponential and trigonometric parts, this opens the possibility to prove the placement of the zeta function on the critical line.