A Rigorous Error Bound for the TG Kernel in Prime Counting

Abstract

We establish rigorous error bounds for prime counting using a truncated Gaussian (TG) kernel in the explicit formula framework. Our main theorem proves that the approximation error remains globally below 1/2 for all sufficiently large arguments, guaranteeing exact computation of {\pi}(x) through simple rounding, without relying on unproven hypotheses. The TG kernel construction employs Gaussian-like test functions with compact support, engineered with vanishing moments to eliminate main terms. For x with 10^8 decimal digits, we demonstrate that only ~1200 nontrivial zeta zeros suffice to achieve the error bound, enabling computation in seconds on modern hardware - a dramatic improvement over classical methods. Key contributions include: (1) Explicit tail truncation bounds using Taylor remainder analysis, showing exponential decay; (2) Zero-sum truncation error bounds via unconditional density estimates; (3) Rigorous treatment of trivial zero contributions. All constants are made explicit, ensuring full verifiability. The method bridges analytic number theory and practical computation, with potential applications to record-breaking prime counting computations. We discuss algorithmic implications including FFT-based arithmetic for ~330 million bit numbers. The framework's flexibility suggests connections to deeper structures in prime distribution, particularly regarding optimized kernel designs and the interplay between smoothing parameters {\alpha} and truncation heights. This work exemplifies how classical analytic techniques, when carefully implemented with modern computational perspectives, yield practical algorithms for problems previously considered purely theoretical. The rigorous error analysis ensures reliability even at astronomical scales, opening new avenues for computational number theory research.