On the Extended, Generalized, and Grand Riemann Hypotheses: A Unified Approach via General Properties of L-Functions

Source abstract

This paper tries to deal with the Extended, Generalized, and Grand Riemann Hypotheses under a unified framework based on the general properties of L-functions. Specifically, the divisibility properties of entire functions expressed as absolutely and uniformly convergent infinite products with irreducible real polynomial factors (as a result of pairing complex conjugate zeros in the Hadamard product), combined with the uniqueness of zero multiplicities and the symmetric functional equation, force all zeros of the completed L-functions 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's methods and conclusions.