Pointwise and correlation bounds on Dedekind sums over small subgroups

We obtain new bounds, pointwisely and on average, for Dedekind sums $\mathsf{s}(\lambda,p)$ modulo a prime $p$ with $\lambda$ of small multiplicative order $d$ modulo $p$. Assuming the infinitude of Mersenne primes, the range of our results is optimal. Moreover, we relate high moments of $L(1,\chi)$ over subgroups of characters to some correlations of Dedekind sums and use our recent results to study these correlations.