Sums of minima and maxima

Let h(1),...,h(n) be positive integers. We study new sums m(h(1),...,h(n)) = Sigma(r1=0)(h1-1) (...) Sigma(rn=0)(hn-1) min {r(1)/h(1),...,r(n)/h(n)} and M(h(1),...,h(n)) = Sigma(r1=0)(h1-1) (...) Sigma(rn=0)(hn-1) max {r(1)/h(1),...r(n)/h(n)}, the first of which times h(1) (...) h(n) is the number of lattice points in a pyramid of dimension n + 1. We show that m(h(1),...,h(n))/(h(1)-1)(...)(h(n)-1) = 1 + Sigma(0not equallsubset of or equal to{1,...,n}) (-1)(\l\) m({h(i)}(iis an element ofl))/Pi(iis an element ofl)(h(i)-1) if h(1),...h(n) > 1, and that M(h(1),...,h(n)) - h(1)...h(n)+1/(h(1)+1)(...)(h(n)+1) = Sigma(0not equallsubset of or equal to{1,...n}) (-1)(\l\) M({h(i)}(iis an element ofl))/Pi(iis an element ofl)(h(i)+1). The sums m(h(1),h(2)) and M(h(1),h(2)) are connected with the reciprocity law for Dedekind sums. The values of m(h(1),h(2),h(3)), M(h(1),h(2),h(3)) and m(h(1),h(2),h(3),h(4)) + M(h(1),h(2),h(3),h(4)) are determined explicitly in the paper. (C) 2002 Elsevier Science B.V. All rights reserved.