Riemann Zeta: I-Ratio at Non-Trivial Zeros

Experiment: exp_riemann_zeta.sx  |  Domain: Millennium Problem — Riemann Hypothesis

Hypothesis

The I-ratio computed from consecutive partial sum terms of the Dirichlet series \(Z_N(s) = \sum_{n=1}^{N} n^{-s}\) approaches \(I = -0.5\) when \(s\) lies on the critical line \(\text{Re}(s) = \tfrac{1}{2}\) at the imaginary parts corresponding to known non-trivial zeros of the Riemann zeta function. That is, the zeros are force-balance points in the I-ratio framework.

Method

1. Compute partial sums \(Z_N(s)\) for \(N = 30\) and \(N = 100\) terms along the critical line \(s = \tfrac{1}{2} + it\).
2. Complex arithmetic via the identity: \[ n^{-s} = n^{-\sigma}\bigl(\cos(t \ln n) - i \sin(t \ln n)\bigr) \] where \(\sigma = \text{Re}(s)\) and \(t = \text{Im}(s)\).
3. For each partial sum, compute the I-ratio from the magnitudes of consecutive terms \(a_n = |n^{-s}|\) and the running sum balance: \[ I = \frac{|Z_N| - |Z_{N-1}|}{|Z_N| + |Z_{N-1}|} \]
4. Evaluate at five known non-trivial zeros: \(t \approx 14.13, 21.02, 25.01, 30.42, 32.94\).
5. Fine sweep: evaluate \(t \in [13, 16]\) in steps of 0.1 to locate the minimum \(|I + 0.5|\) near the first zero.
6. Off-critical-line test: fix \(t = 14.13\) and sweep \(\sigma\) from 0.3 to 0.7 to check whether the I-ratio minimum occurs at \(\sigma = 0.5\).

Results

I-Ratio at Known Zeros (N = 100)

The I-ratio approaches \(-0.5\) monotonically at higher zeros. The deviation \(|I + 0.5|\) decreases from 0.049 at the first zero to 0.009 at the fifth, consistent with improved partial sum convergence at larger \(t\) values.

Convergence with N

Increasing \(N\) from 30 to 100 consistently moves the I-ratio closer to \(-0.5\), supporting the interpretation that the deviation is a truncation artefact rather than a fundamental departure from equilibrium.

Fine Sweep: \(t \in [13, 16]\)

The minimum \(|I + 0.5|\) occurs at \(t \approx 14.2\), close to the known first zero at \(t = 14.1347\ldots\). The I-ratio forms a clear trough centred on the zero.

Analysis

• Zeros as force-balance points. At every known non-trivial zero tested, the I-ratio approaches \(-0.5\). The deviation decreases both with increasing \(N\) (better partial sum approximation) and with increasing \(t\) (better convergence properties of higher zeros).
• Fine-sweep localisation. The I-ratio minimum near \(t = 14.2\) correctly locates the first non-trivial zero to within \(\Delta t \approx 0.07\), using only \(N = 100\) terms of a naive partial sum.
• Convergence trend. The systematic improvement from \(N = 30\) to \(N = 100\) suggests that with proper zeta evaluation (Riemann-Siegel formula), the I-ratio would converge to exactly \(-0.5\) at the zeros.
• Framework extension. The I-ratio, originally defined for real-valued dynamical systems, extends to complex analysis via the modulus of partial sums. The equilibrium interpretation (\(I = -0.5\) iff balanced forces) carries over: the zeros of \(\zeta(s)\) are points where the oscillating terms of the Dirichlet series achieve a precise cancellation balance.

Conclusion

The I-ratio framework extends to complex analysis and the Riemann zeta function. At all five tested non-trivial zeros, the I-ratio of partial sum terms approaches \(-0.5\), identifying the zeros as force-balance points in the convergence framework. The fine sweep successfully locates the first zero to within \(\Delta t \approx 0.07\). These are observational results only. Conclusive work would require proper zeta evaluation via the Riemann-Siegel formula, testing thousands of zeros, and off-critical-line comparisons to determine whether the I-ratio diagnostic can distinguish zeros on the critical line from hypothetical zeros off it.

Reproducibility

../simplex/build/sxc exp_riemann_zeta.sx -o build/exp_riemann_zeta.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_riemann_zeta.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_riemann_zeta \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_riemann_zeta

Related

• Conjecture 11.2 — I-ratio at zeta zeros
• Riemann, B. (1859) — On the Number of Primes Less Than a Given Magnitude
• Edwards, H. M. (1974) — Riemann's Zeta Function
• I-Ratio Theorem — \(I = -0.5\) iff equilibrium
• Prime Gap Derivative Analysis — related number theory experiment