On the p-adic valuation of harmonic numbers
On the p-adic valuation of harmonic numbers
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DOI：
10.1016/j.jnt.2016.02.020
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发表时间：
2016-09
影响因子：
0.7
通讯作者：
C. Sanna
中科院分区：
文献类型：
--
作者：
C. Sanna
对于任意素数p，设J p是正整数n的集合，使得p能整除n次调和数H n的分子。Eswarathasan和Levine的一个旧猜想指出J p是有限的。证明了当x≥ 1时，Jp ∈ [1，x]中的整数个数小于129 p2/3x0.765.特别地，J p具有渐近密度零。此外，我们证明了存在一个对数密度大于0.273的正整数子集Sp，使得对任意n∈ Sp，Hn的p-adic赋值等于−。
For any prime number p, let J p be the set of positive integers n such that p divides the numerator of the n-th harmonic number H n. An old conjecture of Eswarathasan and Levine states that J p is finite. We prove that for x≥ 1 the number of integers in J p∩[1, x] is less than 129 p 2/3 x 0.765. In particular, J p has asymptotic density zero. Furthermore, we show that there exists a subset S p of the positive integers, with logarithmic density greater than 0.273, and such that for any n∈ S p the p-adic valuation of H n is equal to−⌊ log p n⌋.