Congruences involving alternating multiple harmonic sum

TL;DR

This paper proves a congruence involving an alternating sum and harmonic sums modulo prime powers, expanding understanding of harmonic sum identities in number theory.

Contribution

It establishes a new congruence relating an alternating sum with harmonic sums modulo p^3, using alternating multiple harmonic sums.

Findings

The sum involving alternating binomial coefficients is congruent to a harmonic sum modulo p^3.

The proof expresses the sum as a combination of alternating multiple harmonic sums.

The result holds for all odd primes p.

Abstract

We show that for any prime prime $p\not=2$ $$\sum_{k=1}^{p-1} {(-1)^k\over k}{-{1\over 2} \choose k} \equiv -\sum_{k=1}^{(p-1)/2}{1\over k} \pmod{p^3}$$ by expressing the l.h.s. as a combination of alternating multiple harmonic sums.