Ramanujan's Sums and Cyclotomic Polynomials
Ramanujan's Sums and Cyclotomic Polynomials
复制标题
拉马努金的和与分圆多项式
DOI：
--
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
K. Motose
中科院分区：
文献类型：
--
作者：
K. Motose
本文由三个部分组成。首先，我们将证明一些关于Ramanujan和和割圆多项式的公式等价于割圆多项式的定义。接下来，我们还将证明对于x≥32，割圆多项式是严格递增的。最后证明了某些经典级数的值与S在zeta函数的逆1ζ(S+1)中的前三个系数是等价的。设Γn是复数域C中n次单位根的循环群，∆n是本原n次单位根的集合。然后我们有Γn={ζn|0≤k≤n−1}和∆n={ζn|(k，n)=1，1≤k≤n}，其中ζn=e 2πin n，(k，n)是k和n的最大公约数。按顺序对Γn的元素进行分类，Γn是∆d与d|n的不交的子集，即Γn=∪
This paper consists of three parts. First, we shall show that some formulas about Ramanujan’s sum and cyclotomic polynomials are equivalent to the definition of cyclotomic polynomials. Next, we shall show also that cyclotomic polynomials are strictly increasing for x ≥ 32 . The last part is to show that values of some classical series are equivalent to first three coefficients of s in the inverse 1 ζ(s+1) of the zeta function. Let Γn be the cyclic group of n th roots of unity in the complex number field C and let ∆n be the set of primitive n th roots of unity. Then we have Γn = {ζ n |0 ≤ k ≤ n − 1} and ∆n = {ζ n | (k, n) = 1, 1 ≤ k ≤ n} where ζn = e 2πi n and (k, n) is the greatest common divisor of k and n. Classifying elements of Γn by the orders, Γn is the disjoint union of subsets ∆d with d|n, namely, Γn = ∪