We present an explicit asymptotic formula for the imaginary parts tₙ of the non-trivial zeros of the Riemann zeta function. The formula expresses tₙ as a correction to the n-th Gram point gₙ, involving a universal constant that remarkably contains the golden ratio φ = (1+√5) /2. Main result: tₙ = gₙ - C / ln (gₙ) + o (1/ln (gₙ) ) where: - gₙ is the n-th Gram point (solution to θ (gₙ) = (n-1) π) - C = 2π - φ² + ln (2) ≈ 4. 358298499 - φ = (1+√5) /2 ≈ 1. 618033989 (golden ratio) The formula has been validated on 8000 zeros spanning 10 orders of magnitude (n = 1 to n = 10¹⁰), with median relative error < 0. 00001% globally. The validation is organized in 8 uniform tranches of 1000 zeros each: Validation summary: - Total zeros tested: 7, 990 (excluding n < 11 where asymptotic formula fails) - Median error: < 0. 00001% - Mean error: 0. 009% - Predictions with error < 0. 05%: 95. 9% - Predictions with error < 1. 0%: 99. 9% - 100% accuracy for n ≥ 10⁴ The appearance of the golden ratio in this context is unexpected and currently unexplained. We invite the mathematical community to investigate the theoretical origin of this constant. Complete Python implementation, validation data (8000 zeros), and convergence analysis are included in the ancillary files.
Frederic Requiere (Thu,) studied this question.