Improving the error term in the sieve of Eratosthenes

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.