NEW INVESTIGATIONS INTO THE ROLE OF GAMMA FUNCTION DERIVATIVES IN PRIME NUMBER THEORY

This paper synthesizes a framework to probe the Riemann Hypothesis (RH) through the L-function (the n-th derivative of the Gamma function evaluated at unity). Profound connections between L(n) and the Riemann zeta function ζ(s) are explored using tools from fractional calculus, integral transforms and asymptotic analysis which offer new integral representations, functional equations and numerical pathways to investigate zeta zeros. Key contributions include: Functional relations: Derivation of recurrence formulas linking L(n) to ζ(s) via differentiation of the Gamma product formula and Euler’s reflection identity; Integral representations: Development of generalized Cauchy-type repeated integration formulas; Special values & zeros: Computation of L(n) at half-integers and identification of complex zeros; hypothesizing and presenting a pathway to find a solution to the RH and other problems. The results of this paper aim to catalyze further research into the role of Gamma function derivatives in prime number theory.