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^(s_n) = n admits the complex solutions s_n = 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^(s_n) = 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