Some arithmetic properties of recurrence sequences (Q750481)

Let \(\{u(x)\}^{\infty}_{x=1}\) be a linear recursive sequence of integers. This sequence is called nondegenerate if none of the roots and none of the quotients of the roots of its characteristic polynomial are roots of unity. Let S be a set of r prime numbers. For integers \(0<M,N\) let A(S;M,N) denote the number of integers \(M\leq t\leq M+N\) such that u(t) is divisible only by elements from S. It is proved that if \(\{u(x)\}^{\infty}_{x=1}\) is nondegenerate then \(A(S;M,N)\ll r(N \ln (M+N)/M+1),\) and \[ (1/N)\sum^{N}_{u(x)\neq 0;x=1}\phi (| u(x)|)/| u(x)| =\gamma +O(\frac{\ln \ln N}{\ln N}), \] where \(\gamma >0\) is a constant and \(\phi\) denotes Euler's function.