Published Paper Details:

: Hypergeometric functions, which generalize elementary functions and transcendental functions, have profound applications in number theory. Their role extends from providing explicit formulas for special values of zeta functions to contributing to the understanding of modular forms, elliptic curves, and the distribution of prime numbers. In particular, the interplay between hypergeometric series and the theory of modular functions offers insights into the properties of quadratic forms, class numbers, and the behavior of L-functions. This paper explores the key applications of hypergeometric functions in number theory, emphasizing their utility in evaluating sums over lattice points, understanding the asymptotics of number-theoretic functions, and their connections to deep results in transcendental number theory. Through a careful analysis, we demonstrate how hypergeometric series facilitate the derivation of identities and the development of computational methods that have become essential tools in modern number theory research.