HYPOTHÈSES DE REIMMAN

Title An Algebraic Approach to Li-Type Contributions and the Critical Line of the Riemann Zeta Function Description — copy/paste into Zenodo This document presents an independent exploratory mathematical study of the Riemann zeta function and its non-trivial zeros. The work develops an algebraic approach to contributions associated with conjugate pairs of hypothetical non-trivial zeros � and �. In particular, it studies the quantity  \Delta_n(\rho) 2-\left(1-\frac1{\rho}\right)^n -\left(1-\frac1{\bar{\rho}}\right)^n,  together with its asymptotic behavior and its transformation under the functional symmetry �. The document also investigates the transformation  q=1-\frac1{\rho},  for which the condition � is equivalent to �. This provides an algebraic representation of the critical line in the �-plane. Several exact algebraic identities, asymptotic expansions, numerical experiments, symmetry relations, and possible connections with Li-type criteria are developed and examined. Particular attention is given to the distinction between exact mathematical identities, numerical observations, conjectural interpretations, and implications that remain unproved. The document also investigates whether the proposed structure could be connected rigorously to a global criterion involving the complete set of non-trivial zeros of the Riemann zeta function. The principal unresolved issue is the passage from finite or individual zero contributions to a rigorous statement concerning the infinite set of non-trivial zeros. This document does not claim to prove the Riemann Hypothesis. It is an independent exploratory mathematical research note intended to present the proposed constructions, calculations, intermediate results, limitations, and possible directions for further investigation and verification. The author welcomes independent mathematical scrutiny, verification of the derivations, identification of counterexamples, and comparison with existing results in analytic number theory and equivalent formulations of the Riemann Hypothesis. Keywords Riemann Hypothesis; Riemann zeta function; non-trivial zeros; critical line; Li criterion; Li coefficients; analytic number theory; number theory; complex analysis; asymptotic analysis; functional equation; zeta zeros; mathematical research; independent research Metadata Creator: Ravanombahoaka Ismaël Resource type: Preprint Language: English Version: 1.0 License: CC BY 4.0