Executive Summary : | The arithmetic and statistics of Fourier coefficients of modular forms has been a central theme in Number theory and Mathematics. Ramanujan's revolutionary paper posed three conjectures on the Fourier coefficients τ(n) of the Ramanujan ∆ function, with the third conjecture being the most difficult. This led to the distribution of normalised Hecke eigenvalues as p → ∞. In the 1960s, Tate and sato predicted that the normalised Hecke eigenvalues were equidistributed with respect to semi circular measure (sato-Tate measure) as p → ∞. This topic, known as the sato-Tate conjecture, has inspired many theories in Mathematics. In 1997, serre studied a vertical version of the conjecture, known as the "vertical sato-Tate conjecture." This theme has been taken over by many mathematicians worldwide and has connections to various branches of Mathematics and science. Applications of this topic include Geometry, Graph theory, Random matrix theory, hyper geometric series, Chao's theory, Representation theory, and many branches of Physics. In 2009, Murty and sinha obtained the rate of convergence in serre's result. The proposed project aims to achieve similar results in arithmetic progression, with more finer applications in intermediate fields, geometry, and graph theory. |