ErgodicPNT

M. Anshel and D. Goldfeld, "Zeta Functions, One-Way Functions, and Pseudorandom Number Generators", Duke Mathematical Journal, Vol. 88 No. 2 (1997) 371-390. "In 1997,Anshel and Goldfeld [6],presented an explicit construction of a pseudorandom number generator arising from an elliptic curve,which can be effectively computed at low computational cost. They introduced a new intractable problem,distinct from integer factorization or the discrete log problem, that leads to a new class of one-way functions based on the theory of zeta functions,and against which there is no known attack."- Richard M. Mollin,"Introduction to Cryptography" CRC Press (2000)

We show that the Mobius function mu(n) is strongly asymptotically orthogonal to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected nilpotent Lie group with a discrete and cocompact subgroup L (so G/L is a nilmanifold), g : Z -> G is a polynomial sequence and F: G/L -> R is a Lipschitz function. More precisely, we show that the inner product of mu(n) with F(g(n)L) over {1,...,N} is bounded by 1/log^A N, for all A > 0. In particular, this implies the Mobius and Nilsequence conjecture MN(s) from our earlier paper ``Linear equations in primes'' for every positive integer s. This is one of two major ingredients in our programme, outlined in that paper, to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collection \psi_1,...,\psi_t : Z^d -> Z of linear forms all take prime values. The proof is a relatively quick application of the results in our recent companion paper on the distribution of polynomial orbits on nilmanifolds. We give some applications of our main theorem. We show, for example, that the Mobius function is uncorrelated with any bracket polynomial. We also obtain a result about the distribution of nilsequences n -> a^nxL as n ranges only over the primes.

I have recently posted on my web page a preprint concerning some averages of “singular series” (another example of pretty bad mathematical terminology…) arising in the prime k-tuple conjecture, and its generalization the Bateman-Horn conjecture. The reason for looking at this is a result of Gallagher which is important in the original version of the proof by Goldston-Pintz-Yildirim that there are infinitely many primes p for which the gap q-p between p and the next prime q is smaller than ε times the average gap, for arbitrary small ε>0.

We verify a very recent conjecture of Farmer and Rhoades on the asymptotic rate of growth of the derivatives of the Riemann xi function at . We give two separate proofs of this result, with the more general method not restricted to . We briefly describe other approaches to our results, give a heuristic argument, and mention supporting numerical evidence.

Random matrix theory has been a hot topic in number theory, particularly since the Rudnick and Sarnak landmark work on the spacing of consecutive zeros of L-functions. This highly accessible talk has a far more elementary flavour, focusing on eigenvalues of random integer matrices instead of the Gaussian Unitary Ensemble. For a fixed n, consider a random n×n integer matrix with entries bounded by the parameter k. I'll give a simple proof that such a matrix almost certainly has no rational eigenvalues (as k increases). Then we'll delve into more detail on the exact eigenvalue distribution of the 2×2 case. Along the way we'll rediscover a forgotten determinant identity and tackle some quadruple sums. This is joint work with Greg Martin.

这里，简单说几句关于中国人合作中的钱和感情的问题。