  We show that there exists a Boolean function $F$ which observes the following
separations among deterministic query complexity $(D(F))$, randomized zero
error query complexity $(R_0(F))$ and randomized one-sided error query
complexity $(R_1(F))$: $R_1(F) = \widetilde{O}(\sqrt{D(F)})$ and
$R_0(F)=\widetilde{O}(D(F))^{3/4}$. This refutes the conjecture made by Saks
and Wigderson that for any Boolean function $f$,
$R_0(f)=\Omega({D(f)})^{0.753..}$. This also shows widest separation between
$R_1(f)$ and $D(f)$ for any Boolean function. The function $F$ was defined by
G{\"{o}}{\"{o}}s, Pitassi and Watson who studied it for showing a separation
between deterministic decision tree complexity and unambiguous
non-deterministic decision tree complexity. Independently of us, Ambainis et al
proved that different variants of the function $F$ certify optimal (quadratic)
separation between $D(f)$ and $R_0(f)$, and polynomial separation between
$R_0(f)$ and $R_1(f)$. Viewed as separation results, our results are subsumed
by those of Ambainis et al. However, while the functions considerd in the work
of Ambainis et al are different variants of $F$, we work with the original
function $F$ itself.
