多项式的多重倒数和与多重倒数星和几乎绝非整数
数论
2018-07-03 v2
摘要
设 n n n k k k 1 ≤ k ≤ n 1\le k\le n 1 ≤ k ≤ n f ( x ) f(x) f ( x ) m m m f ( m ) ≠ 0 f(m)\ne 0 f ( m ) = 0 k k k s ⃗ = ( s 1 , . . . , s k ) \vec{s}=(s_1, ..., s_k) s = ( s 1 , ... , s k ) H k , f ( s ⃗ , n ) : = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n ∏ j = 1 k 1 f ( i j ) s j H_{k,f}(\vec{s}, n):=\sum\limits_{1\leq i_{1}<\cdots<i_{k}\le n} \prod\limits_{j=1}^{k}\frac{1}{f(i_{j})^{s_j}} H k , f ( s , n ) := 1 ≤ i 1 < ⋯ < i k ≤ n ∑ j = 1 ∏ k f ( i j ) s j 1 H k , f ∗ ( s ⃗ , n ) : = ∑ 1 ≤ i 1 ≤ ⋯ ≤ i k ≤ n ∏ j = 1 k 1 f ( i j ) s j . H_{k,f}^*(\vec{s}, n):=\sum\limits_{1\leq i_{1}\leq \cdots\leq i_{k}\le n} \prod\limits_{j=1}^{k}\frac{1}{f(i_{j})^{s_j}}. H k , f ∗ ( s , n ) := 1 ≤ i 1 ≤ ⋯ ≤ i k ≤ n ∑ j = 1 ∏ k f ( i j ) s j 1 . s j s_j s j H k , f ( s ⃗ , n ) : = H k , f ( n ) H_{k,f}(\vec{s}, n):=H_{k,f}(n) H k , f ( s , n ) := H k , f ( n ) H k , f ∗ ( s ⃗ , n ) : = H k , f ∗ ( n ) H_{k,f}^*(\vec{s}, n):=H_{k,f}^*(n) H k , f ∗ ( s , n ) := H k , f ∗ ( n ) n ≥ 4 n\geq 4 n ≥ 4 f ( x ) = a x + b f(x)=ax+b f ( x ) = a x + b a a a b b b H k , f ( n ) H_{k,f}(n) H k , f ( n ) f ( x ) f(x) f ( x ) k k k s ⃗ = ( s 1 , . . . , s k ) \vec{s}=(s_1, ..., s_k) s = ( s 1 , ... , s k ) f ( x ) = x f(x)=x f ( x ) = x H k , f ( s ⃗ , n ) H_{k,f}(\vec{s},n) H k , f ( s , n ) H k , f ∗ ( s ⃗ , n ) H_{k,f}^*(\vec{s},n) H k , f ∗ ( s , n ) f ( x ) f(x) f ( x ) deg f ( x ) ≥ 2 \deg f(x)\ge 2 deg f ( x ) ≥ 2 f ( x ) f(x) f ( x ) 1 ≤ j ≤ k 1\le j\le k 1 ≤ j ≤ k j j j s j ≥ 2 s_j\ge 2 s j ≥ 2 f ( x ) = x m f(x)=x^{m} f ( x ) = x m m ≥ 1 m\geq1 m ≥ 1 n = k = 1 n=k=1 n = k = 1 H k , f ( s ⃗ , n ) H_{k,f}(\vec{s}, n) H k , f ( s , n ) H k , f ∗ ( s ⃗ , n ) H_{k,f}^*(\vec{s}, n) H k , f ∗ ( s , n ) H k , f ( s ⃗ , n ) H_{k,f}(\vec{s}, n) H k , f ( s , n ) H k , f ∗ ( s ⃗ , n ) H_{k,f}^*(\vec{s}, n) H k , f ∗ ( s , n ) f ( x ) = 2 x − 1 f(x)=2x-1 f ( x ) = 2 x − 1 n = 1 n=1 n = 1 H k , f ( s ⃗ , n ) H_{k,f}(\vec{s}, n) H k , f ( s , n ) H k , f ∗ ( s ⃗ , n ) H_{k,f}^*(\vec{s}, n) H k , f ∗ ( s , n ) H k , f ( s ⃗ , n ) H_{k,f}(\vec{s}, n) H k , f ( s , n ) H k , f ∗ ( s ⃗ , n ) H_{k,f}^*(\vec{s}, n) H k , f ∗ ( s , n ) p p p
引用
@article{arxiv.1703.07263,
title = {Multiple reciprocal sums and multiple reciprocal star sums of polynomials are almost never integers},
author = {Shaofang Hong and Liping Yang and Qiuyu Yin and Min Qiu},
journal= {arXiv preprint arXiv:1703.07263},
year = {2018}
}
备注
20 pages