A Generating Function Technique for Beatty Sequences and Other Step Sequences

✍ Scribed by Kevin O'Bryant

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 178 KB
• Volume: 94
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2001.2743

✦ Synopsis

Let g(x, n), with x ¥ R + , be a step function for each n. Assuming certain technical hypotheses, we give a constant a and function f such that ; . n=1 g(x, n) can be written in the form a+; 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 ; . n=1 z Nn/xM =( 1 z -1) ; 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 z(z)=; .

the Lambert series for Euler's totient function, and ; . n=0 (-1) n sz(2n+1) 2n+1 = p 4 z 1+z 2 , where z(z) and z(z, a) are the Riemann and Hurwitz zeta functions and s z (n)=; 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 (N2naM -NnaM) .

n=1 .