Abstract
We give a quick tour through some topics in analytic prime number theory, focusing in particular on the strange mixture of order and chaos in the primes. For instance, while primes do obey some obvious patterns (e.g. they are almost all odd), and have a very regular asymptotic distribution (the prime number theorem), we still do not know a deterministic formula to quickly generate large numbers guaranteed to be prime, or to count even very simple patterns in the primes, such as twin primes p,p+2. Nevertheless, it is still possible in some cases to understand enough of the structure and randomness of the primes to obtain some quite nontrivial results.
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References
Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, PRIMES is in P. Annals of Mathematics (2) 160, 781–793 (2004)
Euclid, The Elements, circa 300 BCE
Great Internet Mersenne Prime Search. http://www.mersenne.org (2008)
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions. Annals of Mathematics 167(2), 481–547 (2008)
Ben Green and Terence Tao, Linear equations in primes. Preprint. http://arxiv.org/abs/math/0606088, 84 pages (April 22, 2008)
Ben Green and Terence Tao, The Möbius function is asymptotically orthogonal to nilsequences. Preprint. http://arxiv.org/abs/0807.1736, 22 pages (April 26, 2010)
Ben Green, Terence Tao and Tamar Ziegler, The inverse conjecture for the Gowers norm. Preprint
Jacques Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bulletin de la Société Mathématique de France 24, 199–220 (1896)
Godfrey H. Hardy and John E. Littlewood, Some problems of ‘partitio numerorum’. III. On the expression of a number as a sum of primes. Acta Mathematica 44, 1–70 (1923)
Gary L. Miller, Riemann’s hypothesis and tests for primality. Journal of Computer and System Sciences 13(3), 300–317 (1976)
Polymath4 project: Deterministic way to find primes. http://michaelnielsen.org/polymath1/index.php?title=Finding_primes
Michael O. Rabin, Probabilistic algorithm for testing primality. Journal of Number Theory 12, 128–138 (1980)
Lowell Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II. Mathematics of Computation 30, 337–360 (1976)
Charles-Jean de la Vallée Poussin, Recherches analytiques de la théorie des nombres premiers. Annales de la Société scientifique de Bruxelles 20, 183–256 (1896)
Ivan M. Vinogradov, The method of trigonometrical sums in the theory of numbers (Russian). Travaux de l’Institut Mathématique Stekloff 10 (1937)
Don Zagier, Newman’s short proof of the prime number theorem. American Mathematical Monthly 104(8), 705–708 (1997)
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© 2011 Springer-Verlag Berlin Heidelberg
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Tao, T. (2011). Structure and Randomness in the Prime Numbers. In: Schleicher, D., Lackmann, M. (eds) An Invitation to Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19533-4_1
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DOI: https://doi.org/10.1007/978-3-642-19533-4_1
Publisher Name: Springer, Berlin, Heidelberg
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