The Explicit Formula for the Chebyshev–Von Mangoldt Function and the Prime Representing Constant

The Prime Representing Constant infinitely generates prime numbers. We discovered how to compute the Prime Representing Constant using the Hadamard product expansion. We also modified the classical Riemann–von Mangoldt explicit formula. While mathematically equivalent to the classical formula, the cosine-phase form is novel in its computational and structural presentation, enabling faster, memory efficient, and intuitive computation of ψ(x). Combining all the properties will give us a trace-type oscillatory operator that infinitely generates prime numbers.