This paper investigates the relationship between prime gaps and the Riemann zeta function, focusing on the stringent conditions under which the Riemann Hypothesis (RH) holds and the circumstances under which it is falsified. Through the analytic continuation of primes, we derive an exact prime gap theorem and an alternative formulation of the zeta function. A key result reveals that the zeta function ��(log �������� + ����) generates infinite number of zeroes outside the critical strip. Other result reveals that a zero is generated independently of ��, providing a potential counterexample to RH. This challenges the assumption that all non-trivial zeta zeros lie on the critical line ℜ(��)=1 2 . Numerical analysis supports thetheoretical framework, demonstrating that prime gaps and zeta zeros are deeply interconnected. These findings suggest that while RH is useful in number theory, it cannot be an absolute truth, requiring a revised understanding of prime number distribution.