Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun, and Li-Lu Zhao. "Arithmetic theory of harmonic numbers (II)." Colloquium Mathematicae 130.1 (2013): 67-78. <http://eudml.org/doc/283502>.

@article{Zhi2013,
abstract = {For k = 1,2,... let $H_k$ denote the harmonic number $∑_\{j=1\}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have $∑_\{k=1\}^\{p-1\} (H_k)/(k2^k) ≡ 7/24 pB_\{p-3\} (mod p²)$, $∑_\{k=1\}^\{p-1\} (H_\{k,2\})/(k2^k) ≡ - 3/8 B_\{p-3\} (mod p)$, and $∑_\{k=1\}^\{p-1\} (H²_\{k,2n\})/(k^\{2n\}) ≡ (\binom\{6n+1\}\{2n-1\} + n)/(6n+1) pB_\{p-1-6n\} (mod p²)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_\{k,m\}: = ∑_\{j=1\}^k 1/(j^m)$.},
author = {Zhi-Wei Sun, Li-Lu Zhao},
journal = {Colloquium Mathematicae},
keywords = {harmonic numbers; congruences; Bernoulli numbers},
language = {eng},
number = {1},
pages = {67-78},
title = {Arithmetic theory of harmonic numbers (II)},
url = {http://eudml.org/doc/283502},
volume = {130},
year = {2013},
}

TY - JOUR
AU - Zhi-Wei Sun
AU - Li-Lu Zhao
TI - Arithmetic theory of harmonic numbers (II)
JO - Colloquium Mathematicae
PY - 2013
VL - 130
IS - 1
SP - 67
EP - 78
AB - For k = 1,2,... let $H_k$ denote the harmonic number $∑_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have $∑_{k=1}^{p-1} (H_k)/(k2^k) ≡ 7/24 pB_{p-3} (mod p²)$, $∑_{k=1}^{p-1} (H_{k,2})/(k2^k) ≡ - 3/8 B_{p-3} (mod p)$, and $∑_{k=1}^{p-1} (H²_{k,2n})/(k^{2n}) ≡ (\binom{6n+1}{2n-1} + n)/(6n+1) pB_{p-1-6n} (mod p²)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k,m}: = ∑_{j=1}^k 1/(j^m)$.
LA - eng
KW - harmonic numbers; congruences; Bernoulli numbers
UR - http://eudml.org/doc/283502
ER -