Novel Finite Closed-Form Formulas for Divisor Functions with Applications to Two-Prime Sums

We introduce scaled root-of-unity sequences to rewrite the Lambert generating functions of thedivisor functions σ(n) and τ (n). Using finite summation over discrete roots of unity, real-part extraction of complex-valued kernels, and rigorous error estimates, we derive novel finite closed-formexpressions for the sum-of-divisors function and divisor-count function. We build two sets of auxiliary sequences modulo 2n and modulo 8, each furnishing equivalent closed-form identities. Theseformulas are applied to analyze the divisor of integers 2n − p where p is prime, and we explore theirimplications for representations of even integers as sums of two primes. All derived identities arevalidated with numerical computations on small positive integers.