An Investigation on the Prime and Twin Prime Number Functions by Periodical Binary Sequences and Symmetrical Runs in a Modified Sieve Procedure

In this work, the Sieve of Eratosthenes procedure (in the following named Sieve procedure) is approached by a novel point of view, which is able to give a justification of the Prime Number Theorem (P.N.T.). Moreover, an extension of this procedure to the case of twin primes is formulated. The proposed investigation, which is named Limited INtervals into PEriodical Sequences (LINPES) relies on a set of binary periodical sequences that are evaluated in limited intervals of the prime characteristic function. These sequences are built by considering the ensemble of deleted (that is, 0) and undeleted (that is, 1) integers in a modified version of the Sieve procedure, in such a way a symmetric succession of runs of zeroes is found in correspondence of the gaps between the undeleted integers in each period. Such a formulation is able to estimate the prime number function in an equivalent way to the logarithmic integral function Li(x). The present analysis is then extended to the twin primes, by taking into account only the runs whose size is two. In this case, the proposed procedure gives an estimation of the twin prime function that is equivalent to the one of the logarithmic integral function Li2(x). As a consequence, a possibility is investigated in order to count the twin primes in the same intervals found for the primes. Being that the bounds of these intervals are given by squares of primes, if such an inference were actually proved, then the twin primes could be estimated up to infinity, by strengthening the conjecture of their never-ending.