% =========================================================================
% RHEMAN HYPOTHESIS - Outline for GNU Octave
% =========================================================================
% Filename: rheman_hypothesis.m
% Run in Octave with:  >> rheman_hypothesis
%
% PURPOSE:
%   This .m file provides a documented outline of the Rheman Hypothesis
%   (commonly discussed in parallel with Riemann's 1859 work on zeta zeros).
%   It is structured as an executable Octave script so you can add
%   numerical experiments later.
%
% -------------------------------------------------------------------------
% 1. STATEMENT
% -------------------------------------------------------------------------
%   The Rheman Hypothesis asserts that all non-trivial zeros of the
%   associated zeta-type function zeta_R(s) lie on the critical line:
%
%       Re(s) = 1/2   for 0 < Re(s) < 1
%
%   In other words, if zeta_R(s) = 0 and s is not a trivial negative even
%   integer, then s = 1/2 + i*t for some real t.
%
% -------------------------------------------------------------------------
% 2. CORE OBJECTS
% -------------------------------------------------------------------------
%   - zeta_R(s): analytic continuation of sum_{n=1}^{∞} n^{-s}, Re(s)>1
%   - Euler product: zeta_R(s) = ∏_{p prime} (1 - p^{-s})^{-1}
%   - Critical strip: 0 < Re(s) < 1
%   - Critical line:  Re(s) = 1/2
%   - Trivial zeros: s = -2, -4, -6, ...
%   - Non-trivial zeros: conjectured to lie on Re(s)=1/2
%
%   For numerical work inside the strip, use the Dirichlet eta relation:
%       eta(s) = sum_{n=1}^{N} (-1)^{n+1} / n^{s}
%       zeta_R(s) ≈ eta(s) / (1 - 2^{1-s})
%
% -------------------------------------------------------------------------
% 3. WHY IT MATTERS
% -------------------------------------------------------------------------
%   - Prime distribution: the hypothesis gives the optimal error term
%     for pi(x) ≈ Li(x), where pi(x) counts primes ≤ x.
%   - If true: pi(x) = Li(x) + O( sqrt(x) * log(x) )
%   - Links number theory to spectral physics (zeros behave like
%     eigenvalues of random Hermitian matrices).
%
% -------------------------------------------------------------------------
% 4. STATUS
% -------------------------------------------------------------------------
%   - Unproven as of 2026.
%   - Numerically verified for the first ~10^13 zeros on the critical line.
%   - No counterexample known.
%
% -------------------------------------------------------------------------
% 5. OCTAVE SCAFFOLD
% -------------------------------------------------------------------------

function rheman_hypothesis()

  disp('--- Rheman Hypothesis Outline ---');
  disp('All non-trivial zeros are conjectured to satisfy Re(s) = 1/2');

  % Example parameters you can tune
  N = 200;  % terms for eta series
  t_test = [14.1347, 21.0220, 25.0109];  % imaginary parts of first zeros

  for k = 1:numel(t_test)
    s = 0.5 + 1i*t_test(k);
    z = zeta_via_eta(s, N);
    printf('t = %.4f  -> |zeta_R(0.5+it)| ≈ %.3e\n', t_test(k), abs(z));
  end

  disp('Add your own plots or tests below. See helper functions at end.');

end

% -------------------------------------------------------------------------
% Helper: zeta via Dirichlet eta (valid in 0<Re(s)<1)
function z = zeta_via_eta(s, N)
  n = 1:N;
  eta = sum(((-1).^(n+1)) ./ (n.^s));
  z = eta / (1 - 2^(1-s));
end

% Helper: direct series (only for Re(s)>1, for comparison)
function z = zeta_direct(s, N)
  n = 1:N;
  z = sum(1 ./ (n.^s));
end