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<y<x. Then: Taking and in the ESM, we obtain, respectively, (2) (3) (for details, see Apostol 55-57) Substituting (1) into (3) then gives us: (4) The Mobius Inversion Formula (MIF) (for proof, see Apostol 40): Suppose f and g are functions of a real variable. Then we have:, where is the Mobius function. Using the MIF with g(x)=1, we obtain (with some effort): (5) With g(x)=x, and using (2), we obtain: (6) Also, applying the MIF to (4) with and gives us: (7) In light of (5) and (6), all three terms are O(x), so we have proven that. The Prime Number Theorem Richard Lu Swarthmore College, Department of Mathematics & Statistics References Apostol, Tom. 1970. Introduction to Analytic Number Theory. Springer- Verlag, New York. Gioia, Anthony. 1973. A Note on Chebyshev’s Theorem. Mathematics Magazine 46,2:95-96. Havil, Julian. 2003. Gamma: Exploring Euler’s Constant. Princeton University Press, Princeton. Jameson, G.J.O. 2003. The Prime Number Theorem. Cambridge University Press, Cambridge. Levinson, Norman. 1969. A Motivated Account of an Elementary Proof of the Prime Number Theorem. The American Mathematical Monthly 76,3:225-245. Shapiro, H.N. 1950. On the number of primes less than or equal to x. Proceedings of the American.Mathematical.Society 1:346-348. University of Tennessee at Martin. 2006. How many primes are there? Accessed 2006 Oct 12. On the WWW, the Swarthmore math department can be found at http://www.swarthmore.edu/NatSci/math_stat/ Proofs of the Prime Number Theorem The prime number theorem was first proved in 1898 by Hadamard and de la Vallee Poussin, using techniques of complex analysis. Some mathematicians theorized the existence of an elementary proof, while others doubted its existence. In particular, Hardy predicted that an elementary proof would “cause the whole theory to be rewritten.” (Jameson 206-207) An elementary proof of the PNT was discovered in 1948 by Selberg and Erdos. While the two mathematicians had originally collaborated, they ended up independently publishing different versions. Since then, other versions of the PNT have appeared, most notably Levinson’s version (Levinson 225-245). Most make use of Selberg’s formula to some degree (an interesting formula in of itself) (Jameson 206-207). Acknowledgements I’d like to thank Prof. Walter Stromquist for his very helpful comments and suggestions. Interpretation of the Prime Number Theorem We can interpret the PNT as stating that the density of the prime numbers less than n is approximately 1/log n. The graph below gives us a sense as to the accuracy of the approximation. Source: http://primes.utm.edu/howmany.shtml We previously stated one version of the PNT as  (n) ~ n/log n. In fact, it has been shown that n/(log n – 1) provides an even better estimate for the prime counting function, as can be seen in the table below. Source: Jameson 3. Note that the inequality  (n) < Li(n) seems to always hold, at least for the values of n we have presented in the table. Surprisingly, Littlewood has shown that in fact  (n) - Li(n) switches signs an infinite number of times! (Havil 199)