EXTENSIONS OF THE RIEMANN HYPOTHESIS IN COMPLEX ANALYSIS

Abstract

The Riemann Hypothesis (RH) stands as one of the most significant unsolved problems in mathematics, positing that all non-trivial zeros of the Riemann zeta function lie on the critical line in the complex plane. Its implications extend deeply into number theory, particularly concerning the distribution of prime numbers. This study explores various extensions of the Riemann Hypothesis within the realm of complex analysis, highlighting their connections to broader mathematical frameworks. The Generalized Riemann Hypothesis (GRH) extends RH to Dirichlet L-functions, asserting that their non-trivial zeros also lie on the critical line, thereby influencing the distribution of primes in arithmetic progressions. Further broadening this scope, the Grand Riemann Hypothesis (GRH) encompasses all automorphic L-functions, linking the zeros of these functions to the Langlands program, which seeks to unify various aspects of number theory and representation theory.