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What is the Prime Number Theorem? The Prime Number Theorem gives an asymptotic answer to the question “How many primes are there less than n (where n is.

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Presentation on theme: "What is the Prime Number Theorem? The Prime Number Theorem gives an asymptotic answer to the question “How many primes are there less than n (where n is."— Presentation transcript:

1 What is the Prime Number Theorem? The Prime Number Theorem gives an asymptotic answer to the question “How many primes are there less than n (where n is some integer)?” That such an asymptotic formula exists is quite interesting, given the seemingly random distribution of the primes. In particular, the PNT states that  (n) ~, or, in an equivalent form,  (n) ~ li(n), where  (n) = the prime counting function. Counts the number of primes less than or equal to some integer, n. Li(x) = dt Note that  (n) ~ li(n) is typically thought of as the more natural estimate. A Proof of Chebyshev’s Theorem The elementary proof of the PNT is too lengthy to fit in the margins of this poster, much less the poster itself. Instead, we will prove Chebyshev’s Thm, a much weaker result, which states that, where and are constants. If, then there must exist a constant such that (Shapiro 347). Therefore, it will be sufficient to prove that. We follow the method enumerated by Gioia (Gioia, 95-96). First, we introduce Mangoldt’s function (Apostol 32), an important function in number theory. For, we have if where p is a prime, and. Otherwise,. Note that since (for proof, see Apostol 32), we have (1) where ( is known as Chebyshev’s function - see Apostol 75) Euler’s Summation Formula (ESM) (for proof, see Apostol 54): Suppose f has a continuous derivative on some interval [y,x], where 0


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