Proto-Zeros of the Truncated Euler Product: A Corrected Numerical and Asymptotic Study

The truncated Euler product ζ_T(s) = ∏_{p≤T} (1 − p^{−s})^{−1} converges to ζ(s) as T → ∞ for Re(s) > 1. We examine the function F_T(t) = 2 Re[−ζ_T′/ζ_T(1/2 + it)], which is real-valued and has zeros at real t. As T increases, these 'proto-zeros' converge toward the imaginary parts γ_n of the Riemann zeta zeros. For T = 4 (primes 2, 3), the proto-zeros occur near t ≈ 11.7, 17.4, 19.8 — not at γ_1 = 14.134…. For T = 1000, they approach the true zeros to within 0.1. The convergence rate appears to be O(1/log T), consistent with the width of the classical zero-free region. The proto-zeros are always real, regardless of whether the true zeros lie on the critical line. This provides a computable, real-valued approximation to the Riemann zeros whose reality is unconditional. The observation does not prove or imply the Riemann Hypothesis.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-07-18
DOI
https://doi.org/10.5281/zenodo.21404196
Primary Topic
Analytic Number Theory Research
Type
preprint
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preprint

Proto-Zeros of the Truncated Euler Product: A Corrected Numerical and Asymptotic Study

Goss, Jr., Matthew J.
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
preprint

Proto-Zeros of the Truncated Euler Product: A Corrected Numerical and Asymptotic Study

Goss, Jr., Matthew J.
preprint en

Abstract

The truncated Euler product ζ_T(s) = ∏_{p≤T} (1 − p^{−s})^{−1} converges to ζ(s) as T → ∞ for Re(s) > 1. We examine the function F_T(t) = 2 Re[−ζ_T′/ζ_T(1/2 + it)], which is real-valued and has zeros at real t. As T increases, these 'proto-zeros' converge toward the imaginary parts γ_n of the Riemann zeta zeros. For T = 4 (primes 2, 3), the proto-zeros occur near t ≈ 11.7, 17.4, 19.8 — not at γ_1 = 14.134…. For T = 1000, they approach the true zeros to within 0.1. The convergence rate appears to be O(1/log T), consistent with the width of the classical zero-free region. The proto-zeros are always real, regardless of whether the true zeros lie on the critical line. This provides a computable, real-valued approximation to the Riemann zeros whose reality is unconditional. The observation does not prove or imply the Riemann Hypothesis.

Zenodo (CERN European Organization for Nuclear Research)
Jupiter Medical Center (US)
Analytic Number Theory Research
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