A Unified Proof of the Extended, Generalized, and Grand Riemann Hypothesis Based on the General Properties of L-Functions

Source abstract

The Extended, Generalized, and Grand Riemann Hypotheses are proved under a unified framework, which is based on the general properties of L-functions. To be specific, the divisibility of entire functions expressed as Hadamard products with irreducible real quadratic polynomial factors (as a result of pairing complex conjugate zeros), combined with the uniqueness of zero multiplicities and the symmetric functional equation, forces all zeros of the entire function in the critical strip onto the critical line. Consequently, the existence of Landau-Siegel zeros is excluded, thereby confirming the Landau-Siegel zeros conjecture. As to the Davenport--Heilbronn counterexample, since it possesses no Euler product---the fundamental structural property that confines zeros to the critical strip, it is not in the scope of this paper.