Harmonic numbers
H
k
=
∑
0
>
j
⩽
k
1
/
j
(
k
=
0
,
1
,
2
,
…
)
H_{k}=\sum _{0>j\leqslant k}1/j\ (k=0,1,2,\ldots )
play important roles in mathematics. In this paper we investigate their arithmetic properties and obtain various basic congruences. Let
p
>
3
p>3
be a prime. We show that
∑
k
=
1
p
−
1
H
k
k
2
k
≡
0
(
m
o
d
p
)
,
∑
k
=
1
p
−
1
H
k
2
≡
2
p
−
2
(
m
o
d
p
2
)
,
∑
k
=
1
p
−
1
H
k
3
≡
6
(
m
o
d
p
)
,
\begin{equation*} \sum _{k=1}^{p-1}\frac {H_{k}}{k2^{k}}\equiv 0\ (\mathrm {mod} \ p),\ \sum _{k=1}^{p-1}H_{k}^{2} \equiv 2p-2\ (\mathrm {mod} \ p^{2}), \ \sum _{k=1}^{p-1}H_{k}^{3}\equiv 6\ (\mathrm {mod} \ p),\end{equation*}
and
∑
k
=
1
p
−
1
H
k
2
k
2
≡
0
(
m
o
d
p
)
provided
p
>
5.
\begin{equation*} \sum _{k=1}^{p-1}\frac {H_{k}^{2}}{k^{2}}\equiv 0\ (\mathrm {mod} \ p)\qquad \text {provided }\ p>5. \end{equation*}
(In contrast, it is known that
∑
k
=
1
∞
H
k
/
(
k
2
k
)
=
π
2
/
12
\sum _{k=1}^{\infty }H_{k}/(k2^{k})=\pi ^{2}/12
and
∑
k
=
1
∞
H
k
2
/
k
2
=
17
π
4
/
360
\sum _{k=1}^{\infty }H_{k}^{2}/k^{2}=17\pi ^{4}/360
.) Our tools include some sophisticated combinatorial identities and properties of Bernoulli numbers.