My paper brings to light new facts about prime numbers, especially about their distribution and approximation using li(x). Although the very definition of a prime number is trivial, it is very difficult to determine exactly how many primes are in a certain interval. Perhaps it is this paradox that is the reason why prime numbers have become the center of number theorists’ attention. From a historical point of view, we know that, after proving the existence of infinitely many primes, mathematicians tried to develop a method or to define a function with which it would be possible to obtain information about how primes are distributed among integers. They called this function Prime counting function π(x). Unfortunately, it was out of their reach to define a function with which they would be able to quickly and only with elementary methods calculate π(x) exactly, especially for large x. For this reason, they tried to define such a continuous function that would at least approximate π(x) as much as possible. One of those who tried to do this was Gauss, who introduced the function called logarithmic integral li(x) almost more than 200 years ago. Although numerical calculations showed a good estimate of the π(x) using li(x), the error term Δ(x) still played its role and especially the question about the behavior of Δ(x), if x → ∞. Gauss even suggested, based on his calculations, that π(x) ~ li(x). Based on this, π(x) was understood as a function that consists of two functions. Namely, π(x) = li(x) + Δ(x). For the next period, especially of the Δ(x)19th century, the aim was no longer to define a more accurate function than li(x), although everyone would have wanted that, but rather to better understand the behavior of Δ(x). A major breakthrough came in 1859 when Riemann introduced his function R(x) that, after neglecting the terms involving complex numbers, revealed a direct relation between π(x) – li(x) for the first time in history. Another breakthrough came in 1896 when the Prime number theorem (PNT) was proven. In other words, PNT proved what Gauss suggested π(x) ~ li(x), and PNT also proved a certain upper bound for Δ(x). In the 20th century, it was proven that Δ(x) changes sign infinitely often, and a sharper bound for |Δ(x)| was defined as implied from PNT. It is natural to continuously improve something and, therefore, in my paper, we will directly follow up on the historical sequence of events and define an even more accurate bound for |Δ(x)| as it has been until now. The paper totals 10 pages, is written very clearly, and does not include elements of complex analysis, which will not lead to a selection of readers who do not have knowledge about it. In the field of number theory, prime numbers are still a hot and debated topic, and almost every interested professional or amateur tries to learn something new about it or contribute to it in some way. I firmly believe that my paper will become a source of new information on which people will be able to buildand develop their new conjectures and theories. At the same time, I will be very happy if my article helps solve any open problems and if it motivates others to study prime numbers, which can be another step forward in understanding and consolidating knowledge about prime numbers.