As we have seen, and proved in many different ways, there are infinitely many primes. In fact, as Dirichlet’s theorem shows, there are infinitely many primes in any arithmetic progression an + b with (a, b) = 1. However, an examination of the list of positive integers shows that the primes become scarcer as the integers increase. This statement was quantified in Theorem 2.3.2, where we proved that there are arbitrarily large spaces or gaps within the sequence of primes. As a result of these observations the question arises concerning the distribution or density of the primes. The interest centers here on the prime number function π(x) defined for positive integers x by π(x) = number of primes ≤ x.
KeywordsAnalytic Continuation Zeta Function Arithmetic Progression Critical Line Elementary Proof
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