Arithmetic theory of harmonic numbers

Abstract

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.