Arithmetic theory of harmonic numbers (II)
By Zhi-Wei Sun, Lilu Zhao. 2009
47 citations recorded by the source corpus.
For $k=1,2,\ldots$ let $H_k$ denote the harmonic number $\sum_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime $p>3$ we have $$\sum_{k=1}^{p-1}\frac{H_k}{k2^k}\equiv\frac7{24}pB_{p-3}\pmod{p^2},\ \ \sum_{k=1}^{p-1}\frac{H_{k,2}}{k2^k}\equiv-\frac 38B_{p-3}\pmod{p},$$ and $$\sum_{k=1}^{p-1}\frac{H_{k,2n}^2}{k^{2n}}\equiv\frac{\binom{6n+1}{2n-1}+n}{6n+1}pB_{p-1-6n}\pmod{p^2}$$ for any positive integer $n<(p-1)/6$, where $B_0,B_1,B_2,\ldots$ are Bernoulli numbers, and $H_{k,m}:=\sum_{j=1}^k 1/j^m$.
Subfields: Algebra and Number Theory, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/b5432aa509d4bd4ec66515a378c66a1f74e1718f
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In Algebra and Number Theory
In Statistical and Nonlinear Physics
Citations in the corpus
- Open Conjectures on Congruences (2009, background): “jecture B8 (raised in [S09h], and confirmed by the author’s former student Hui-Qin Cao). If p is a prime with p ≡ 11 (mod 12), then pX−1 k=0 k 3 2k k 2 (−16)k ≡ 0 (mod p). Conjecture B9 (raised in [S09g], and confirmed by the author’s former student Li-Lu Zhao (Hong Kong University)). Let m be any p”