This work develops a unified operational toolbox for the systematic integration of a broad class of integrals appearing in analysis, analytic number theory, and mathematical physics. The guiding principle is spectral diagonalization: integration is equivalent to moving the problem to a domain where the kernel becomes algebraic, operating there, and returning with a closed expression. Three classical tools are reinterpreted as diagonalization operators: (1) the Heaviside operational calculus for polynomial-exponential, trigonometric, and hyperbolic integrands; (2) the Riemann–Liouville fractional integral for Abel-type singular kernels and powers; and (3) the Mellin transform as an operational logarithm for logarithmic weights and multiplicative structures. These three tools are unified by the spectral operator , defined on Laplace–Mellin-type representations, which acts, depending on the parameter , as ordinary differentiation , fractional integration , or logarithmic weight insertion . The Alpha family1 over primes appears as a spectral orbit of under the semigroup . It includes a rigorous axiomatic framework of reduction, a structured taxonomy of definite and improper integrals (Gamma, Beta, trigonometric powers, rational oscillatory integrals), and a section on the precise limits of the toolbox, respecting the Liouville–Risch theorem 1The Alpha family over primes is not originally introduced in this manuscript. Its systematic development belongs to the independent work of Ramón Moya, Theory of Alpha Functions on Primes: Fundamentals, Analytic Properties and Functional Equations of Quotient (Zenodo, 2026, DOI: 10.5281/zenodo.19298685), where the Alpha functions on primes, their analytical properties, their fundamental differential system, their integral representations, and their quotient functional equations are developed. If the reader wishes to review the definition used in this work, they may consult §6.8, Definition 6.4.
Ramón Moya (Sat,) studied this question.