DRCC-Width Zeta Method (DWZ-V2)

Abstract

We study a controlled, finite-width approximation of the Riemann zeta function in the critical strip, the region between the boundary lines on which the real part of the complex variable equals zero and one. Our starting point is the DRCC stability condition, Dimensional Reduction via Controlled Combinatorics, due to Hesamiy, requiring a reconstruction depth that remains positive, finite, and bounded above by a suitable fragment size. The simplest conceivable approximation, the sum truncated after a fixed number of series terms, fails this condition inside the critical strip: it diverges as the truncation depth grows, rather than approaching the true zeta function. We resolve this with a first-order correction based on the classical Euler-Maclaurin summation formula, a standard technique of analysis known since the eighteenth century and not a discovery of this work. The corrected approximation provably converges to the true zeta function, including inside the critical strip. We deliberately use the simplest version of this correction as our main model, because it is the first member of the correction family at which convergence occurs at all, not because it is fastest. A comparison with stronger corrections and with the established Riemann-Siegel formula instead shows a clear efficiency disadvantage of two to three orders of magnitude. From the observed convergence behavior we derive a simple rule of thumb for how deeply the series must be truncated to reach a desired accuracy. The method was checked against one hundred known zeros of the zeta function using high-precision reference values, with deviations consistently in the range of a few millionths to ten-millionths, two found blindly. A direct runtime comparison shows the established Riemann-Siegel formula to be considerably faster and more accurate, a result reported openly. This work claims neither a new Euler-Maclaurin formula, nor a proof of the Riemann Hypothesis, nor a replacement for established methods, but a mathematically transparent, reproducible framework for the controlled approximation of the zeta function.

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