Improving the error term in the sieve of Eratosthenes

Abstract: We have devised an alternative approach to sifting integers in the sieve of Eratosthenes that helps refine the error term. Instead of eliminating all multiples of a prime number $p<z$ in the traditional sieve method, our approach solely eliminates multiples of $p$ that have the minimum prime factor of $p$. By leveraging the density of integers with the least prime factor $p$ in this sieve technique, we obtain a reduced error term and an upper bound of $\pi(x)$ that accurately reflects the prime number theorem.