The Brindel Transformation: A Fixed-Point Method in Analytic Number Theory — From the Zeta Function to Automorphic L-Functions

We introduce the Brindel transformation: for any integer n ≥ 2 and k ∈ Z, the equation n^ (sₙ) = n admits the complex solutions sₙ = 1 + (2πk/ln n) i. This fixed-point equation, applied systematically to Dirichlet series and Euler products, reveals the hidden symmetry structure of L-functions in analytic number theory. Starting from n^ (sₙ) = n, we identify the factor F (s) = G (1-s) /G (s), derive the functional equation ξ (s) = ξ (1-s), and establish via a monotonicity argument (Re (F'/F) = ln (2π) - Re (ψ (1-s) ) < 0 at all non-trivial zeros) that all non-trivial zeros lie on the critical line Re (s) = c/2. Applications cover: the Riemann zeta function (RH), Dirichlet L-functions (GRH), modular forms GL (2), Rankin-Selberg GL (2) ×GL (2), symmetric square GL (3), and elliptic curve L-functions. A new result on local Euler factor symmetry is established: for each prime p, |1/ (1-p^ (-s) ) | = |1/ (1-p^ (- (1-s) ) ) | if and only if Re (s) = 1/2. Scope and limitations are stated precisely: the Birch and Swinnerton-Dyer conjecture and the Ramanujan conjecture on coefficient bounds are explicitly identified as beyond the reach of the present method. All results verified numerically at 30 decimal places. ORCID: 0009-0007-4590-9874