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

Pointwise and correlation bounds on Dedekind sums over small subgroups