Spectral and Arithmetic Perspectives on Generalized Zeta-like Functions: Selberg Class, Modular Forms, and Quantum Chaos

This proposal introduces a unified spectral framework for understanding zeta-like functions through Fourier structures. We demonstrate how classical objects such as the Riemann zeta function, Dirichlet L-functions, and modular forms can be embedded into a spectral function space G∞\mathcal{G}_\inftyG∞ via logarithmic frequency transforms. The proposal further explores connections between the zero distributions of these functions and quantum chaotic systems, emphasizing the role of the Selberg trace formula, Maass waveforms, and modular symmetries. This spectral reinterpretation bridges number theory and quantum physics, suggesting new directions in arithmetic geometry and spectral analysis.