Sieve Procedure for the M\"obius 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 M\"obius prime-function, which in the limit of infinitely large primes becomes identical to the original M\"obius 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.Comment: 16 pages, critically reviewed both proofs of the Prime Number Theorem, added details and new result

On the Fourier expansion method for highly accurate computation of the Voigt/complex error function in a rapid algorithm

In our recent publication [1] we presented an exponential series approximation suitable for highly accurate computation of the complex error function in a rapid algorithm. In this Short Communication we describe how a simplified representation of the proposed complex error function approximation makes possible further algorithmic optimization resulting in a considerable computational acceleration without compromise on accuracy.Comment: 4 page