The Densities Of Primes Represented By Some Quadrat

The distribution of primes is an ancient problem in number theory,Euclid proved that there are infinite primes three century B.C. with the elementary methods. Dirich-let introduced L-function and Dirichlet density when he studied the distribution of primes in the arithmetic series. Dirichlet density is the limit of the ratio which is the L-function of some prime ideals satisfied some properties and function-log(s-1). Using Dirichlet density, Dirichlet proved that primes are uniformly distributed in arith-metic series without any limit, so there are infinite primes in arithmetic series. We can compute the densities of primes represented by some quadratic forms using ring class fields and Cebotarev density theorem. Furthermore, we can consider the density of primes represented by some quadratic forms in arithmetic series. In this paper we will study the primes represented by some special quadratic forms in the arithmetic series with common difference p,where p is a prime. We will compute their Dirichlet densitues and also find that they are uniformly distributed in arithmetic series. We will give a application of them on Vandiver’s conjectures.In the first chapter, we will introduce preliminary of the primes represented by quadratic forms, and give some basic notion such as quadratic forms, order in quadratic fields, class group and class number, ring class field, Dirichlet density. We will construct the isomorphism of the class group of the quadratic forms and the class group of the corresponding order in a imaginary quadratic field, and find the formula of their class number. We will construct ring class fields using class field theory, discuss their prop-erties. At last we will give the formula of Dirichlet densities of the primes represented by quadratic forms.In the second chapter,we will mainly discuss the distribution of the primes rep-resented by some special quadratic forms in arithmetic series. Firstly we will consider the equations q=x2+p2n+1y2.Restricting q in the arithmetic series with common d-ifference p, using ring class fields and Cebotarev density theorem we can compute their Dirichlet densities. Secondly we will consider the equations 4q=x2+p2n+1y2. We will also use ring class fields and Cebotarev density theorem to compute their Dirichlet densities. Similarly, restricting q in the arithmetic series with common difference p, we can compute their Dirichlet densities. By the results of the above densities, we can find the equation 4q=x2+p2n+1y2 have no odd solution on some conditions. At last we will consider the equations gh= x2+p2n+1y2 and 4qh= x2+p2n+1y2,where h is the class number of Q((?)), we will compute their Dirichlet densities.In the third chapter, we will give a application of the Dirichlet densities in the second chapter. Vandiver’s conjectures says that the p-part of the ideal class group of Q(ζp+ζp-1) is trivial. Using the Thaine’s results we can make a nontrivial component e(p+1)/2(A) of them due to the quadratic forms with the special divisible properties. But by the Dirichlet densities in the second chapter we can show such quadratic forms do not exist, so we get that the component e(p+1)/2(A) is trivial.