A Smooth Continuous Analytic Construction of the n-th Prime Number

We present a novel construction of a smooth, continuous, and purely analytic function that outputs the n^th prime number, without the use of conditional logic, modular arithmetic, floor or ceiling functions, or any piecewise definitions. This function is composed entirely of integrals, exponential functions, and summations, tools from continuous mathematics, yet it succeeds in isolating discrete prime values in exact form. The construction begins by defining a prime detector function P(n), which evaluates to 1 if n is prime and 0 otherwise, using only exponential sums and a single definite integral. From this, we define the prime-counting function C(x) as the sum of P(j) for j≤x, and use an integral-based identity to filter the unique integer k for which C(k)=n and P(k)=1. The result is a symbolic expression for p_n, the n^th prime number, expressed entirely in analytic terms. To our knowledge, this constitutes the first such construction with these properties. Our work demonstrates a new pathway for bridging continuous and discrete mathematics and opens potential for future analytic formulations of arithmetic functions.