Arithmetic theory of harmonic numbers (II)

• Autorzy: Zhi-Wei Sun , Li-Lu Zhao
• Języki publikacji: EN

Abstrakty

EN

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 $H_{k,m}: = ∑_{j=1}^k 1/(j^m)$.

Kategorie tematyczne

• 11B75: Other combinatorial number theory
• 05A19: Combinatorial identities, bijective combinatorics
• 11B68: Bernoulli and Euler numbers and polynomials
• 11A07: Congruences; primitive roots; residue systems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2013
• Tom: 130
• Numer: 1
• Strony: 67-78

Daty

wydano

2013

Twórcy

autor

Zhi-Wei Sun

• Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China

autor

Li-Lu Zhao

• School of Mathematics, Hefei University of Technology, Hefei 230009, People's Republic of China

Identyfikatory

DOI

10.4064/cm130-1-7