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Arithmetic theory of harmonic numbers (II)

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Sun Z.-W., Zhao L.-L.

Colloquium Mathematicum

2013

tom 130

nr 1

67-78

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 $H_{k,m}: = ∑_{j=1}^k 1/(j^m)$.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Sun Z.-W.

1 Zhao L.-L.

1 2013

Wyniki wyszukiwania

Arithmetic theory of harmonic numbers (II)