On The Problem For Number Of Solutions Of Binary Polynomial Congruences

Let f(x)=xd+a1xd-1+…+ad,a1,...,ad∈Z,d≥2, be an irreducible polynomial. Let Nf(n) be the number of solutions x of f(x)=0(mod n) satisfying0≤x≤n. It is an important problem to study the function Nf(n).Early in1952, Erdos did some research on this problem, and got two asymptotic formulas. In recent years. many mathematicians around the world have done much research on the number of solutions of congruences. Fomenko, Kim and Lu have done some papers on the number of solutions of certain congruences and have got good results.In1999, Daniel published a paper entitled as On the divisor-sum problem for binary forms. In his paper, he got an estimation of the number of solutions of binary polynomial congruences. In this paper, I do some further research on this problem, make the estimation much better and get two generalized results.This paper is mainly composed of three parts. The first part introduces systematically the background of the subject and gives the results:Let f(x1,x2) be an irreducible binary polynomial with integer coefficients of degree k (k≥2), and let (?)(a),(?)(a) be as follows:Theorem1.1For any Abelian binary polynomial f(x1,x2) of degree k, we have where C(f) is defined in (3.1.4).Theorem1.2For any non-Abelian binary polynomial f(x1,x2) of degree k (k≥9), we have where C(f) is defined in (3.2.2).Theorem1.3For any l≥2, where Pm(logQ) is defined in (3.3.5), m=kl-1.Theorem1.4For any l≥2, where Pm’(logQ) is defined in (3.4.3), m=kl-1.The second part introduces the preliminary knowledge useful to prove the theorems, including the estimation of some power integral mean value of ζ(s); L(s,x) in analytic number theory, the properties of Dedekind zeta-function in algebraic number theory, Gabriel’s convexity theorem. Phragmcn Lindelof theorem, the properties of binary polynomials and so on.The third part firstly recalls some properties of function (?)(a) and (?)*(a), and then separately gives the proofs of Theorem1.1, Theorem1.2, Theorem1.3and Theorem1.4. The proof in this paper applies many methods and skills of analytic number theory, and uses Perron’s formula, Cauchy’s residue theorem and the properties of Dcdckind zcta-function to get the final estimatio...