一類三角和數的表示式及其近似估值

<正> §1.本文的主要目的是要證明下面的結果: 定理1.設q_1,q_2為二正整數.合

A NOTE ON THE EXPRESSIONS AND ESTIMATIONS OF A KIND OF TRIGONOMETRICAL SUMS

The object of this note is to prove the following theorems.Theorem 1. Let q_1 and q_2 be two integers, and Let then where H_1~((1))(ω) and H_1~((2))(ω) are the Hankel functions of first and second kind respectively andK_1(ω) is the modified Hankel function.As direct consequences of this theorem, we have the following corollaries:Corollary 1. (Hardy, )Corollary 2. Let q_1q_2 > 1. If q_1=1, then set h=0, (similarly for q_2). Then Corollary 3. Let X_1(n) and X_2(n) be two characters of modulus q and r respectively, then where υ is the Euler constant, τ(X_1) and the terms in the great brackets correspond to (i) X_1(n) and X_2(n) are principal characters; (ii) X_1(n) is a principal character and X_2(n) is a non-principal character; (iii) X_1(n) is a non-principal character and X_2(n) is a principal character; (iv) X_1(n) and X_2(n) are non-principal characters.Corollary 4. (Hardy) where U(n) is the number of integral solutions of u~2+υ~2=n in u, υ; J_1(ω) is the Bessel function.Corollary 5. Let K be a quadrat field with integral diseriminant d, then the number of integral ideals a in K with morm Na < ω~2 is equal to where a is the number of unit roots, ε_1 is the fundamental unit, h is the number of classes of K.By means of a known method,we can proveTheorem 2.As theorem 1, this theorem has the same applications.Corollary 6.Corollary 7. with the same principal terms as (1).