A new type of inclusion exclusion principle for sequences and asymptotic formulas for \(\zeta(k)\)
Let \(\{a(n)\}\) be a sequence of integers defined by a nondegenerate binary linear recurrence. It is shown that there is a constant \(\kappa\) (depending on the recurrence) such that the following asymptotic formula holds for every positive integer \(l\): \[ \log {| a(1^ l)a(2^ l) \cdots a(n^ l) | \over [a(1^ l),a(2^ l),\dots,a(n^ l)]} ={\zeta(l+1) \over 1-\kappa}+O \left( {\omega(n) \over n} \right), \] where square brackets denote the LCM, \(\zeta(s)\) is the Riemann zeta function, \(\omega(n)=\log n\) for \(l=1\), and \(\omega (n)=1\) for \(l\geq 2\). This improves and extends an earlier result by the author [J. Number Theory 36, 328-331 (1990; Zbl 0708.11016)] for the case \(l=1\).
scientific article; zbMATH DE number 484043