SmallRatio: Dedekind sums along Euclid's algorithm and the small-ratio formula
For a prime p and 0 = 1 with uv < p/2, |12 s(h,p) - p sum ([hv = u mod p] - [hv = -u mod p])/(uv)| <= 6n <= 12 log_2 p + 6, where n is the length of Euclid's algorithm on (p, h). Summed over any set H of residues, in particular over a subgroup of odd order, this gives |12 sum_{h in H} s(h,p) - p S_H(p)| <= |H| (12 log_2 p + 6), where S_H(p) is the signed sum of 1/(uv) over the coprime pairs with uv < p/2 and u/v in H or -H. The development proves Dedekind reciprocity from scratch (by counting lattice points), the identities of Euclid's algorithm and the bound n <= 2 log_2 k + 1, the…
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