An extension of Carlson's theorem for entire functions of exponential type (Q914885)

For an entire function \(f=u+iv\) of exponential type less than \(\pi\), satisfying \(\sum | f(n)| <\infty\) (where n runs through the integers), the conditions \[ u(n)=v(ne^{i\alpha})=0 \] for all positive integers n imply f(z)\(\equiv 0\) provided that \(\alpha\) is real and exp(2 im \(\alpha\))\(\neq -1\) for all positive integers m.