  In the work of S. Petermichl, S. Treil and A. Volberg it was explicitly
constructed that the Riesz transforms in any dimension $n \geq 2$ can be
obtained as an average of dyadic Haar shifts provided that an integral is
nonzero. It was shown in the paper that when $n=2$, the integral is indeed
nonzero (negative) but for $n \geq 3$ the nonzero property remains unsolved. In
this paper we show that the integral is nonzero (negative) for $n=3$. The
novelty in our proof is the delicate decompositions of the integral for which
we can either find their closed forms or prove an upper bound.
