Deriving the Error Term Bound Coefficient in the Prime Number Theorem via a Sieve-Based Algorithm

The distribution of prime numbers has long been a central topic in analytic number theory. The Prime Number Theorem (PNT), which states that a number of primes less than x is \( x/log(x) \), provides a foundational understanding of this phenomenon. The further study of deep insights has led to the Riemann Hypothesis (RH), which implies explicit bounds on the error term in the PNT, thus enhancing the precision of the results derived from it. In this work, an algorithm is proposed for the estimation of the prime-counting function by counting the number of composite numbers eliminated during the sieving of the odd-number sequence. By applying this approach, it was found that variations in the length of the odd-number sequence during removal of composite numbers follow an oscillation pattern governed by the sinc function. Further analysis of such oscillations suggests that the function \( π(x) \) is composed of two terms: the main term, which makes the largest contribution to the distribution of the primes, and the error term, which is responsible for the accumulation of errors during the calculation of the \( \text{sinc} \) function values up to the limit \( m \), defined as \( √x /2 \). This leads to the proposal that the bound coefficient of the error term should be equal to \( √x/log x \), which is correlated with an estimate of this coefficient derived under the assumption of the truth of the RH. We hope that this perspective stimulates future work toward formalizing this approach to uncover deeper connections to the zeta function and prime number theory at large scale.