Of Minima and Maxima

Let h1, · · · , hn be positive integers. We study new sums m(h1, · · · , hn) = h1−1 ∑ r1=0 · · · hn−1 ∑ rn=0 min { r1 h1 , · · · , rn hn } and M(h1, · · · , hn) = h1−1 ∑ r1=0 · · · hn−1 ∑ rn=0 max { r1 h1 , · · · , rn hn } , the first of which times h1 · · ·hn is the number of lattice points in a pyramid of dimension n + 1. We show that m(h1, · · · , hn) (h1 − 1) · · · (hn − 1) = 1 + ∑ ∅6 =I⊆{1,··· ,n} (−1)|I| m({hi}i∈I) ∏ i∈I(hi − 1) if h1, · · · , hn > 1, and that M(h1, · · · , hn)− h1 · · ·hn + 1 (h1 + 1) · · · (hn + 1) = ∑ ∅6 =I⊆{1,··· ,n} (−1)|I| M({hi}i∈I) ∏ i∈I(hi + 1) . The sums m(h1, h2) and M(h1, h2) are closely connected with the reciprocity law for Dedekind sums. The values of m(h1, h2, h3), M(h1, h2, h3) and m(h1, h2, h3, h4) + M(h1, h2, h3, h4) are determined explicitly in the paper.