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Arithmetic theory of harmonic numbers (II)

Zhi-Wei SunLi-Lu Zhao — 2013

Colloquium Mathematicae

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 ) / ( k 2 k ) 7 / 24 p B p - 3 ( m o d p ² ) , k = 1 p - 1 ( H k , 2 ) / ( k 2 k ) - 3 / 8 B p - 3 ( m o d p ) , and k = 1 p - 1 ( H ² k , 2 n ) / ( k 2 n ) ( 6 n + 1 2 n - 1 + n ) / ( 6 n + 1 ) p B p - 1 - 6 n ( m o d p ² ) 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 ) .

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