INTEGER POINTS UNDER THE GRAPH OF A CONTINUOUS FUNCTION f (x): TOPOLOGICAL MOSAICS AND ASYMPTOTIC APPLICATIONS
Ключевые слова:
integer points, fractional parts, topological mosaic, Dirichlet problem, analytic number theory, Euler-Maclaurin formula, hypergeometric bounds, asymptotic evaluation, lattice enumeration, combinatorial geometryАннотация
Analyzing integer coordinates within continuous boundaries remains a fundamental
challenge in modern combinatorial geometry and analytic number theory. In this current study, we precisely
compute the exact cardinality and asymptotic distribution of lattice nodes located beneath transcendental
and algebraic curves. Transitioning from discrete enumeration to continuous analytical methods, we utilize
topological loci of fractional parts, Euler-Maclaurin expansions, Fourier series, and hypergeometric bounds
to establish a robust evaluative framework. Our results yield exact analytical identities, proving a strict
bijective correspondence between lattice configurations, and successfully resolving complex asymptotic
error margins in the classical Dirichlet’s divisor problem.
