A generating function technique for Beatty sequences and other step sequences

Let g(x, n), with x is an element of R+, be a step function for each n, Assuming certain technical hypotheses, we give a constant a and function f such that Sigma(n=1)(infinity) g(x, n) can be written in the form alpha + Sigma(0<r<x) f(r), where the summation is extended over all points in (0, x) at which some g((.), n) is not continuous. A typical example is Sigma(n=1)(infinity) z([n/x]) = (1/z - 1) Sigma z(q)/(1 - z(q)), with the summation extending over all pairs p, q of positive integers satisfying 0 < p/q < x and gcd(p, q) = 1. We then apply such representations to prove identities such as zeta(z) = Sigma(n=1)(infinity) phi(n)/n(z) (zeta(z) - zeta(z, 1 + 1/n)), the Lambert series for Euler's totient function, and Sigma(n=0)(infinity) (-1)(n) sigma.(2n+1)/2n+1 = pi/4 z/1+z2, where zeta(z) and zeta(z, a) are the Riemann and Hurwitz zeta functions and sigma(z)(n) = Sigma(d\n) dz(d). We also give a generalization of the Rayleigh Beatty theorem and a new result of a similar nature for the sequences ([2nalpha] - [nalpha])(n=1)(infinity). (C) 2002 Elsevier Science (USA).