Values of the Euler Φ-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
Values of the Euler Φ-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
复制标题
不能被给定奇素数整除的欧拉 Φ 函数的值，以及分圆域的欧拉-克罗内克常数的分布
作者：
Kevin Ford;F. Luca;P. Moree
设φ表示欧拉的φ函数。对于一个固定的奇素数q，我们研究了n × 6 x的渐近级数展开式的一阶和二阶项，使得q ∈ N（n）.部分分析涉及到对分圆域的欧拉-克罗内克常数的仔细研究。特别是，我们表明，扩展的黎曼假设，总理k-元组猜想和猜想Ihara关于这些欧拉-克罗内克常数的分布不可能都是真的。
Let ϕ denote Euler’s phi function. For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n 6 x such that q ∤ ϕ(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Extended Riemann Hypothesis, the prime k-tuples conjecture and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be all true.