PRIME NUMBER THEOREM Meaning and
Definition
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The prime number theorem is a fundamental result in number theory that describes the asymptotic distribution of prime numbers among the positive integers. It was conjectured by Gauss in the early 19th century and was later proved independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896.
In essence, the prime number theorem states that the number of primes not exceeding a given positive integer x is approximately equal to x divided by the natural logarithm of x. This result provides a mathematical formula to estimate the density of primes within a range of positive integers.
More precisely, the prime number theorem states that the function π(x), which represents the number of primes less than or equal to x, is approximately equal to x/log(x), where log denotes the natural logarithm. This approximation becomes increasingly accurate as x grows larger. In other words, the prime number theorem shows that prime numbers become less frequent as the positive integers increase.
The theorem has profound implications in number theory and has paved the way for further investigations into the distribution of prime numbers. It forms a cornerstone in the study of prime numbers and has inspired numerous other results and conjectures related to prime number distribution, such as the Riemann hypothesis. The prime number theorem stands as one of the most important and celebrated results in mathematics.
