Arithmetic theory of harmonic numbers (II)

Details Arithmetic theory of harmonic numbers (II) Arithmetic theory of harmonic numbers (II)

2009-11-23

arxiv.org/abs/0911.4433v7

Mathematics

Number Theory

Only keep part (II) since part (I) has been published in Proc. AMS 140(2012), 415-428

Scientific paper

For $k=0,1,2,...$ 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 $$\sum_{k=1}^{p-1}H_k/(k2^k)=7/24*pB_{p-3} (mod p^2)$$ and $$\sum_{k=1}^{p-1}H_{k,2}/(k2^k)=-3B_{p-3}/8 (mod p^2),$$ and also $$\sum_{k=1}^{p-1}H_{k,2n}^2/k^{2n}=(\binom{6n+1}{2n-1}+n)/(6n+1)*pB_{p-1-6n} (mod p^2)$$ for any positive integer $n<(p-1)/6$, where $B_0,B_1,B_2,...$ are Bernoulli numbers, and $H_{k,m}:=\sum_{j=1}^k 1/j^m$.

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