Research Famous Formula Arithmetic Functions And Number Theory Some Nature

Researches on the arithmetic properties of number-theoretic functions and distribution of prime numbers have always been important objects in number theory, particularly in Analytic number theory. Dirichlet L-functions, Gauss sums, Dedekind sums and Dirichlet Character sums have glorious history in number theory. They have been researched extensively and many remarkable results are achieved. With the development and further research in number theory, the study on distribution of primes has always been an active research content in number theory.Based on our interests in the above problems, we use analytic and elemen-tary methods to study the arithmetical properties of Dirichlet L-function, Gauss sums, Dedekind sums, Dirichlet Character sums and the distribution of special kind of primes, obtain some interesting results. Concretely, the main achieve-ments contained in this dissertation are as follows:1. Researches on the mean value of Dirichlet L-functions. Let p＞2be a prime, and k≥1be an integer. Let χ be a Dirichlet character modulo p, and let L(s,χ) be the Dirichlet L-function corresponding to χ. We consider the mean value and get the exact formula for it.2. Researches on the hybrid mean value related to Dedekind sums. Using the properties of character sum and the analytic method to study one new mean value related to the Dedekind sums, and give two interesting mean asymptotic formula for it.3. Researches on the distribution of the polynomials of the Dirichlet Char-acter. Let m, n be integers, and k be a positive integer, q=p1α1p2α2…psαs be a square-full number. Let Xi be an even Dirichlet primitive character mod piαi (i=1,2,…, s). Using Gauss sums、Kloosterman sums and the properties of character sums, we study the following value distribution of Dirichlet Character of polynomials χ(mxk+nyk): where pi|q satisfying pi≡1(mod2k) and (mn, q)=1, a denotes the solution of congruence equation ax≡1(mod q), and give an exact formule for it.4. Researches on the distribution of prime numbers of a special type. By Fourier series, Bombieri-Vinogradov theorem and linear sieve, we prove that for any irrational α∈R\Q, β∈R, and0＜θ＜2/375, there are infinitely many primes p such that p+2=p4(p4has at most four prime factors)and‖αp2+β‖＜P-θ.5. Researches on arithmetic properties of the polynomials and special num-bers. By some basic properties of the second kind Stirling numbers, we study the cycle relation of Bernoulli and Euler polynomials, and obtain two closed formula.