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Music of the Primes in Search of Order

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PDF generated at: Wed, 19 Jan 2011 04:12:59 UTC Contents Articles Prime number theorem 1 Riemann hypothesis 9 Riemann zeta function 30 Balanced prime 40 Bell number 41 Carol number 46 Centered decagonal number 47 Centered heptagonal number 48 Centered square number 49 Centered triangular number 51 Chen prime 52 Circular prime 53 Cousin prime 54 Cuban prime 55 Cullen number 56 Dihedral prime 57 Dirichlet's theorem on arithmetic progressions 58 Double factorial 61 Double Mersenne prime 75 Eisenstein prime 76 Emirp 78 Euclid number 78 Even number 79 Factorial prime 82 Fermat number 83 Fibonacci prime 90 Fortunate prime 91 Full reptend prime 92 Gaussian integer 94 Genocchi number 97 Goldbach's conjecture 98 Good prime 102 Happy number 103 Higgs prime 108 Highly cototient number 109 Illegal prime 110 Irregular prime 113 Kynea number 114 Leyland number 115 List of prime numbers 116 Lucas number 131 Lucky number 133 Markov number 135 Mersenne prime 137 Mills' constant 145 Minimal prime (recreational mathematics) 146 Motzkin number 147 Newman–Shanks–Williams prime 149 Odd number 150 Padovan sequence 153 Palindromic prime 157 Partition (number theory) 158 Pell number 166 Permutable prime 174 Perrin number 175 Pierpont prime 178 Pillai prime 179 Prime gap 180 Prime quadruplet 185 Prime triplet 187 Prime-counting function 188 Primeval prime 194 Primorial prime 196 Probable prime 197 Proth number 198 Pseudoprime 199 Pythagorean prime 200 Ramanujan prime 200 Regular prime 202 Repunit 203 Safe prime 208 Self number 209 Sexy prime 212 Smarandache–Wellin number 214 Solinas prime 215 Sophie Germain prime 215 Star number 217 Stern prime 218 Strobogrammatic prime 219 Strong prime 220 Super-prime 222 Supersingular prime (moonshine theory) 223 Thabit number 224 Truncatable prime 225 Twin prime 226 Two-sided prime 229 Ulam number 230 Unique prime 232 Wagstaff prime 234 Wall-Sun-Sun prime 235 Wedderburn-Etherington number 237 Wieferich pair 237 Wieferich prime 238 Wilson prime 242 Wolstenholme prime 243 Woodall number 246 References Article Sources and Contributors 248 Image Sources, Licenses and Contributors 253 Article Licenses License 254 Prime number theorem 1 Prime number theorem In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a rough description of how the primes are distributed. Roughly speaking, the prime number theorem states that if a random number nearby some large number N is selected, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime. In other words, the average gap between prime numbers near N is roughly ln(N).[1] Statement of the theorem Let π(x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1, which is expressed by the formula Graph comparing π(x) (red), x / ln x (green) and Li(x) (blue) known as the asymptotic law of distribution of prime numbers. Using asymptotic notation this result can be restated as This notation (and the theorem) does not say anything about the limit of the difference of the two functions as x approaches infinity. (Indeed, the behavior of this difference is very complicated and related to the Riemann hypothesis.) Instead, the theorem states that x/ln(x) approximates π(x) in the sense that the relative error of this approximation approaches 0 as x approaches infinity. The prime number theorem is equivalent to the statement that the nth prime number p is approximately equal to n n ln(n), again with the relative error of this approximation approaching 0 as n approaches infinity. Prime number theorem 2 History of the asymptotic law of distribution of prime numbers and its proof Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in 1796 that π(x) is approximated by the function x/(ln(x)-B), where B=1.08... is a constant close to 1. Carl Friedrich Gauss considered the same question and, based on the computational evidence available to him and on some heuristic reasoning, he came up with his own approximating function, the logarithmic integral li(x), although he did not publish his results. Both Legendre's and Gauss's formulas imply the same conjectured asymptotic equivalence of π(x) and x / ln(x) stated above, although it turned out that Gauss's approximation is considerably better if one considers the differences instead of quotients. In two papers from 1848 and 1850, the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(s) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π(x)/(x/ln(x)) as x goes to infinity exists at all, then it is necessarily equal to one.[2] He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near to 1 for all x.[3] Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(x) were strong enough for him to prove Bertrand's postulate that there exists a prime number between n and 2n for any integer n ≥ 2. Without doubt, the single most significant paper concerning the distribution of prime numbers was Riemann's 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject. Riemann introduced Distribution of primes up to revolutionary ideas into the subject, the chief of them being that the distribution of prime 19# (9699690). numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π(x) originates. Extending these deep ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Hadamard and de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0.[4] During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, including the "elementary" proofs of Atle Selberg and Paul Erdős (1949). While the original proofs of Hadamard and de la Vallée-Poussin are long and elaborate, and later proofs have introduced various simplifications through the use of Tauberian theorems but remained difficult to digest, a surprisingly short proof [5] [6] was discovered in 1980 by American mathematician Donald J. Newman. Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis. Proof methodology In a lecture on prime numbers for a general audience, Fields medalist Terence Tao described one approach to proving the prime number theorem in poetic terms: listening to the "music" of the primes. We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. Then we prove, and this is the hard part, that certain "notes" cannot occur in this music. This exclusion of certain Prime number theorem 3 notes leads to the statement of the prime number theorem. According to Tao, this proof yields much deeper insights into the distribution of the primes than the "elementary" proofs discussed below.[7] Proof sketch Here is a sketch of the proof referred to in Tao's lecture mentioned above. Like most proofs of the PNT, it starts out by reformulating the problem in terms of a less intuitive, but better-behaved, prime-counting function. The idea is to count the primes (or a related set such as the set of prime powers) with weights to arrive at a function with smoother asymptotic behavior. The most common such generalized counting function is the Chebyshev function , defined by Here the summation is over all prime powers up to x. This is sometimes written as , where is the von Mangoldt function, namely It is now relatively easy to check that the PNT is equivalent to the claim that . Indeed, this follows from the easy estimates and (using big O notation) for any ε > 0, The next step is to find a useful representation for . Let be the Riemann zeta function. It can be shown that is related to the von Mangoldt function , and hence to , via the relation A delicate analysis of this equation and related properties of the zeta function, using the Mellin transform and Perron's formula, shows that for non-integer x the equation holds, where the sum is over all zeros (trivial and non-trivial) of the zeta function.
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