n http mathworld wolfram comPrime Fo rmulas html

소수의 생성 n 소수공식 http: //mathworld. wolfram. com/Prime. Fo rmulas. html

Mersenne 소수 Mersenne numbers 2 n – 1 n 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, . . . n 38번 n=6972593, 2098960 자리수, 1999 년발견 Hajratwala, Woltman, Kurowskiet. al. (GIMPS, Prime. Net) n 완전수와의 관계 n

소수의 분포 가우스, Legendre 예상 n Prime Number Theorem (Hadamard, Poussin) 리만가설 사용 n Erdos-Selberg Elementary Proof using Selberg identity p x (log p)2 + pq x log p log q = 2 x log x + O(x) n

소수의 분포와 양자 혼돈 n "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers. . . grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision. "

Random matrices and Riemann Zeta functions n "For thirty years there have been conjectured connections, supported by ever mounting evidence, between the zeros of the Riemann zeta function and eigenvalues of random matrices. One of the most famous unsolved problems in mathematics is the Riemann hypothesis, which states that all the non-trivial zeros of the zeta function lie on a vertical line in the complex plane, called the critical line. The connection with random matrix theory is that it is believed that high up on this critical line the local correlations of the zeros of the Riemann zeta function, as well as other Lfunctions, are the same as those of the phases of the eigenvalues of unitary matrices of large dimension taken at random from the CUE ensemble of random matrix theory. More recently, however, it was realized that random matrix theory not only describes with high accuracy the distribution of the zeros of L-functions, but it is also extremely successful in predicting the structure of various average values of L-functions that previous number theoretic techniques had not been able to tackle.