A prime number is â€œa natural number greater than 1 that can be divided evenly only by 1 and itselfâ€ (Alfeld). There are great many of prime numbers, the smallest one is 1 and the largest known prime number on the moment of webpage creation is m39Â = 213,466,917-1 (Alfeld). A prime counting function Ï€(x) â€œcounts the number of primes that are less than or equalâ€ to x (Bose 292). Considering the prime counting function Ï€(x), it may be concluded that the value of Ï€(x) increases only when x encounters the prime number. Thus, the function Ï€(x) can be characterized as monotonically increasing one.
Defining the prime number theorem, it is significant to notice that it was conjectured by Legendre and Gauss. To the word, they worked on the theorem separately and managed to achieve the same results independently. A prime number theorem is the statement that â€œthe relative error in the approximation Ï€(x) ~ Li(x) approaches zero as x â†’ âˆžâ€ (Edwards 68). The following statement can be illustrated as follows,
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