Arithmetic Partitions of Natural Numbers, Zeta Regularization, Prime Divisibility and Dirichlet L-Functions

The partition of the positive integers into arithmetic progressions provides a useful framework for studying divergent series through analytic continuation. This paper investigates arithmetic partitions of the natural numbers and their relationship with the Riemann zeta function, Hurwitz zeta function, Bernoulli polynomials, prime divisibility and Dirichlet L-functions. The ordinary series of positive integers diverges, but its zeta-regularized value is ζ(−1) = −1/12. We derive formulas for regularized sums of residue classes modulo an integer, examine partitions determined by prime divisibility, and obtain weighted identities involving Dirichlet characters. We extend the method to higher powers and discuss an infinite partition based on powers of two. Particular attention is given to the distinction between finite additivity and countable additivity in regularized summation.