Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun; Li-Lu Zhao

Zhi-Wei Sun ; Li-Lu Zhao

Colloquium Mathematicae, Tome 131 (2013), p. 67-78 / Harvested from The Polish Digital Mathematics Library

Résumé

For k = 1,2,... 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 $\sum_{k = 1}^{p - 1} (H_{k}) / (k 2^{k}) \equiv 7 / 24 p B_{p - 3} (m o d p ²)$ , $\sum_{k = 1}^{p - 1} (H_{k, 2}) / (k 2^{k}) \equiv - 3 / 8 B_{p - 3} (m o d p)$ , and $\sum_{k = 1}^{p - 1} (H ²_{k, 2 n}) / (k^{2 n}) \equiv ((\binom{6 n + 1}{2 n - 1}) + n) / (6 n + 1) p B_{p - 1 - 6 n} (m o d p ²)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k, m} : = \sum_{j = 1}^{k} 1 / (j^{m})$ .

Publié le : 2013-01-01

Zbl 1290.11052

EUDML-ID : urn:eudml:doc:283502

@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm130-1-7,
     author = {Zhi-Wei Sun and Li-Lu Zhao},
     title = {Arithmetic theory of harmonic numbers (II)},
     journal = {Colloquium Mathematicae},
     volume = {131},
     year = {2013},
     pages = {67-78},
     zbl = {1290.11052},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm130-1-7}
}

Zhi-Wei Sun; Li-Lu Zhao. Arithmetic theory of harmonic numbers (II). Colloquium Mathematicae, Tome 131 (2013) pp. 67-78. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm130-1-7/