Multiple harmonic sums and Wolstenholme's theorem

Abstract

We give a family of congruences for the binomial coefficients $\left(\genfrac{}{}{0ex}{}{kp-1}{p-1}\right)$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$) involves a linear combination of $n$ multiple harmonic sums, and holds $\mathrm{mod}{p}^{2n+3}$. The coefficients in these congruences are integers depending on $n$ and $k$, but independent of $p$. More generally, we construct a family of congruences for $\left(\genfrac{}{}{0ex}{}{kp-1}{p-1}\right)\mathrm{mod}{p}^{2n+3}$, whose members contain a variable number of terms, and show that in this family there is a unique "optimized" congruence involving the fewest terms. The special case $k=2$ and $n=0$ recovers Wolstenholme's theorem $\left(\genfrac{}{}{0ex}{}{2p-1}{p-1}\right)\equiv 1\mathrm{mod}{p}^3$, valid for all primes $p\ge 5$. We also characterize those triples $(n,k,p)$ for which the optimized congruence holds modulo an extra power of $p$: they are precisely those with either $p$ dividing the numerator of the Bernoulli number $B_{p-2n-k}$, or $k\equiv 0,1\mathrm{mod}p$.

22 pages

References in corpus (1)

• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)

Cited by in corpus (6)

• Asymptotic relations for weighted finite multiple zeta values
• The MHS algebra and supercongruences
• Values of symmetric polynomials and a truncated analogue of the Riemann zeta function
• Généralisation des congruences de Wolstenholme et de Morley
• Wolstenholme Type Congruences and Framing of Rational 2-Functions
• $p$-adic L-functions and Classical Congruences