Critical Line in the Euler–riemann Zeta Function - Frank Trefoily* (Annals of Mathematics and Physics, 2025)
Abstract This study focused on the transformation of an exponentially growing divergent function sin(Rln(x)) into a convergent function by its complementary exponential function xt in such a manner that the sizes of positive and negative areas under sin would be the same. The transformation will provide the entire sin function with self-compensatory behavior. The exponent's value was computed and found to be -1/2, which is the only exponent, which lets entire product of the function converge to zero (sum of area for positive real numbers and sum of products for natural numbers). The exponent -1/2 is algebraically and geometrically inevitable for the function xtsin(Rln(x)) converging to zero. This result directly impacts the position of the critical line in the Euler-Riemann zeta function.