98 research outputs found
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 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
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