A Square Root Parity Classification of Primes

This paper introduces a new, geometry-based way of classifying prime numbers using the value of the square root of each prime. Instead of grouping primes by their remainders modulo some number (the usual algebraic approach), this work examines which perfect-square interval a prime falls into. By assigning each prime a “square-root parity” based on the parity of ⌊√p⌋, the paper shows that primes naturally split into two balanced classes that alternate at every perfect square. The study develops this idea into a full analytic framework. It proves that both classes contain exactly half of all primes (in a Dirichlet density sense), and establishes a sharp error bound that controls how far the two classes can differ up to a given number x. The paper also constructs a new Dirichlet-like L-function based on this classification and shows that it admits an Euler product and a clean factorization involving the Riemann zeta function. The results combine classical analytic number theory with a novel geometric viewpoint. They show that even a simple geometric rule looking at which square interval a prime belongs to—produces rich, structured behavior. This suggests that geometric classifications of primes can complement traditional algebraic methods and may lead to new insights about prime distribution, prime constellations, and related analytic questions.