Sums of minima and maxima Original Research Article

Let h1,…,hn be positive integers. We study new sumsm(h1,…,hn)=∑r1=0h1−1 ⋯∑rn=0hn−1 minr1h1,…,rnhnandM(h1,…,hn)=∑r1=0h1−1 ⋯∑rn=0hn−1 maxr1h1,…,rnhn,the first of which times h1⋯hn is the number of lattice points in a pyramid of dimension n+1. We show thatm(h1,…,hn)(h1−1)⋯(hn−1)=1+∑∅≠I⊆{1,…,n} (−1)|I| m({hi}i∈I)∏i∈I (hi−1)if h1,…,hn>1, and thatM(h1,…,hn)−h1⋯hn+1(h1+1)⋯(hn+1)=∑∅≠I⊆{1,…,n} (−1)|I|M({hi}i∈I)∏i∈I (hi+1).The sums m(h1,h2) and M(h1,h2) are 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.