On the asymptotic formula in the fourth-degree estermann problem with almost equal terms: Theory and numerical validation

Abstract

This study investigates a class of number-theoretic problems related to representations of large integers as combinations of prime numbers and quartic terms, extending ideas originating from the Estermann problem. The authors focus on a refined ternary decomposition involving almost equal components, employing analytic methods from additive number theory. Central to the analysis are estimates of trigonometric sums of Weyl type and their applications to bounding error terms in asymptotic formulas. By developing sharper bounds for short exponential sums, the paper establishes a new asymptotic formula for the fourth-degree analogue under constrained conditions. To bridge theory and computation, we further introduce a numerical framework based on Gegenbauer polynomial expansions, which efficiently approximates the associated singular series and exponential integrals with spectral accuracy. The proposed algorithm leverages discrete orthogonality and Gaussian quadrature to achieve high-precision validation of the asymptotic predictions. The obtained results improve previously known bounds, provide rigorous numerical verification, and contribute to a deeper understanding of additive structures involving primes and polynomial terms. These findings have potential implications for related problems in analytic number theory and the development of spectral methods for arithmetic functions.