$1^{-1}+2^{-1}+\dots+\Big(\frac{p-1}{2}\Big)^{-1} \equiv -\frac{2^p - 2}{p} \mod p$ for an odd prime $p.$ I've reduced a problem down to proving this identity. Unfortunately, I don't know where to even start. There has to be some way of expanding the RHS or combining terms on the LHS, but I don't see it. Any hints?

David W. Boyd writes:

There is a relationship between values of certain $H_n$ and Fermat's quotient $q_a = {(a^{p-1}-1)}/p \bmod p$. For example, a result from Eisenstein from 1850 [Dickson 1952, p. 41] states that $H_{(p-1)/2} \equiv -2q_2 \mod p$; this is easily seen from the binomial expansion of ${(1 + 1)}^p$.

This is your hint, and if that's not enough, the reference where to find a proof.
[Dickson 1952]: L. E. Dickson, History of the Theory of Numbers.
[Boyd 1994]: David W. Boyd, A $p$-adic Study of the Partial Sums of the Harmonic Series.
