Riemann Hypothesis And The Distribution Of Primes

Besides giving a survey on Riemann Hypothesis(RH), affine orbits, sat-uration number, affine linear sieve and some conjectures and results about the distribution of primes, this article mainly gives some new conjectures and results about Q(x,k) and the kernel of integers, i.e. conjecture1.8,1.9and theorem1.30in chapter1, and theorem3.2,3.3,3.8and3.9in chapter3. RH is still not proved by now. The non-trivial zeroes of Riemann zeta function contain a lot of information about primes. Additionally, RH has a number of equivalent statements, concerning the realm of elementary number theo-ry, analysis, algebra and ergodic theory and so on. Finally, we can deduce numerous results under RH, and some equivalences have been proved uncon-ditionally, so RH is highly significant. In chapter1, we discuss RH and its equivalences and some results under RH. At last, we give some conjectures about Q(x,k). Because RH is related to the primes, chapter2is about the distribution of primes, such as Dirichlet’s Theorem and Schinzel’s conjecture H and so on. Moreover, we can generalize these questions to one question. But we pay an attention to affine linear sieve. In last chapter, firstly, we sim-ply introduce prime number theory. Finally, we gain some new results on the kernel of integers making use of fundamental and analytic method.