A Gaussian-Perron Smoothed Prime-Side Defect for Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function

Summary · qwen2.5:32b

Researchers introduced a Gaussian-Perron prime-force defect that compares smoothed prime-side forces with the logarithmic derivative of the Riemann zeta function, providing a local diagnostic tool for zero geometry. This construction leads to an exact formula and a universal profile for zeros on the logarithmic scale under the assumption of the Riemann Hypothesis and specific pole-damping conditions. The method incorporates an error-function prime weight and anisotropic damping law to distinguish amplified from suppressed nonlocal contributions near critical-line zeros.

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Exploring a new Gaussian-Perron method to better understand the Riemann Zeta Function and its zero geometry, with potential implications for real-time rendering in computer graphics.

Excerpt

arXiv:2607.04316v1 Announce Type: new Abstract: We introduce a Gaussian--Perron prime-force defect that compares a smoothed prime-side logarithmic force with the logarithmic derivative of the Riemann zeta function. The construction turns the explicit formula into a local diagnostic for zero geometry. Its kernel produces an error-function prime weight and an anisotropic zero-side damping law, with an explicit boundary separating amplified and suppressed nonlocal zero contributions. We prove an exact zero-side formula, derive a universal selected-zero profile on the logarithmic scale, and formulate a finite-window damping certificate for non-selected residues. Under explicit damping, pole, and contour-regularity hypotheses, these ingredients localize the full defect near a selected zero. Assuming the Riemann Hypothesis and the stated pole-damping condition, the full defect near each fixed simple critical-line zero has the selected-zero profile up to an exponentially small nonlocal remainder. The framework provides a local diagnostic for zero geometry associated with the Riemann zeta function.