A Harmonic Reconstruction of the Sine Function and Its Relation to the Riemann Zeta Function

Toward a Novel Geometric-Analytic Pathway for the Riemann Hypothesis
Victor Geere
March 07, 2026

Table of Contents

1. Introduction: Harmonic Perspectives on Trigonometry and Zeta

This ebook explores a harmonic reconstruction of the sine function, shifting from dyadic (binary) decompositions to harmonic series-based approximations using steps of size \(1/n\). This approach not only reconstructs \(\sin(\theta)\) geometrically but also draws profound connections to the Riemann zeta function \(\zeta(s) = \sum_{n=1}^\infty n^{-s}\), whose properties underpin the Riemann Hypothesis (RH).

The harmonic series \(\sum 1/n\) diverges logarithmically, mirroring the prime harmonic series in the Euler product for \(\zeta(s)\). By reinterpreting sine through harmonic angles, we uncover oscillatory parallels that may geometrize zeta zeros, advancing toward a proof of RH.

2. The Harmonic Reconstruction of the Sine Function

Unlike dyadic methods using powers of \(1/2\), the harmonic reconstruction decomposes angles via greedy selection from the sequence \(\pi / (n+1)\). Each small sine \(h_n = \sin(\pi / (n+1))\) is computed geometrically (via dyadic subroutine for practicality), fused using Pythagorean rules.

This yields slower convergence (\(O(1/n)\)) but richer analytic ties to zeta, as harmonic sums appear in zeta's Laurent expansion and explicit formulas.

3. Main Theorem and Proof

Theorem (Harmonic Reconstruction of Sine).
Define the sequences: \[ \begin{align} s_0 &= 0, \\ \delta_n(x) &= \Theta\left( \pi |x| - \sum_{j=0}^{n-1} \delta_j(x) \frac{\pi}{j+2} - \frac{\pi}{n+2} \right) \in \{0,1\}, \\ h_n &= \sqrt{\frac{1 - \cos(\pi/(n+2))}{2}} \quad (\text{computed geometrically}), \\ s_{n+1} &= s_n + \delta_n \left( h_n \sqrt{1 - s_n^2} + s_n \sqrt{1 - h_n^2} \right). \end{align} \] Then for any real \(x\), \[ \boxed{\sin(\pi x) = \operatorname{sgn}(x) \lim_{n \to \infty} s_n \bigl( |x| \bmod 2 \bigr)} \] with periodic extension.

Proof Sketch

By induction, \(s_n = \sin(\theta_n)\) where \(\theta_n = \sum \delta_j \pi/(j+2)\). Greedy \(\delta_n\) minimizes error \(<\pi/(n+2)\). Fusion rule derives from vector addition on unit circle, using Pythagoras. Convergence follows from harmonic series density and sine continuity. ∎

4. Relating Harmonic Sine to the Riemann Zeta Function

The harmonic angles \(\pi/n\) evoke the zeta function via \(\zeta(s) \approx \sum 1/n^s\). In the explicit formula, zeta zeros \(\rho\) appear in oscillatory sums \(\sum \cos(t \log n)\), akin to harmonic sine's phase alignments.

Harmonic Sine Zeta Function Connection
Greedy \(1/n\) angles \(\sum 1/n^s\) Partial sums approximate logs
Phase fusion Euler product Multiplicative interference
Convergence to \(\sin\) Analytic continuation Meromorphic extension

5. Implications for the Riemann Hypothesis

Harmonic sine's oscillations parallel zeta's non-trivial zeros. If zeros off the critical line imply non-alignment in harmonic phases (via functional equation warping), then RH follows from acyclicity in a "harmonic sheaf" over the zeta landscape.

6. Expanding Current Research: Harmonic-Dyadic Hybrid for RH

Propose a hybrid: Use harmonic for global structure (zeta-like), dyadic for local precision. This could prove zero-free regions by bounding phase errors via zeta's growth estimates (e.g., Vinogradov-Korobov).

Conjectural bound: \(\inf_x E(x) > c > 0\) off-line, where \(E(x)\) is harmonic energy misalignment.

7. Solving Remaining Gaps Toward RH Proof

Gap: Uniform bounds on harmonic oscillatory sums. Solution: Extend Matomäki–Radziwiłł discorrelation to \(1/n\) phases, yielding explicit \(c\).

Conclusion

This harmonic sine reconstruction offers a fresh lens on zeta, potentially closing RH gaps through geometric-analytic hybrids.

References