Rigorous Proof and Physical Applications of Analytic Continuation of Fractal Zeta and Implications of Fractal Riemann Hypothesis

This paper aims to rigorously prove the analytic continuation of the fractal Zeta function and explore its applications in fields such as quantum gravity. By constructing an arithmetic representation of self-similar fractal manifolds, we define the fractal Zeta function and demonstrate the existence of its analytic continuation using Mellin transforms and the fractal Gamma function. Furthermore, we propose the fractal Riemann Hypothesis and validate its correctness through the example of the Sierpi´ nski gasket. Finally, we discuss the application of the fractal Zeta function in calculating thermodynamic entropy in quantum gravity and provide perspectives on future research directions.