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Sieve Procedure for the Möbius prime-functions, the Infinitude of Primes and the Prime Number Theorem

Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding Möbius prime-function, which in the limit of infinitely large primes becomes identical to the original Möbius function. Discussing this limit we present two simple proofs of the Prime Number Theorem. In the framework of this approach we give several proofs of the infinitude of primes.