  Let $\lfloor t \rfloor$ denote the greatest positive integer less than or
equal to a given positive real number $t$ and $\vartheta(t)$ the Chebyshev
$\vartheta$-function. In this paper, we prove a certain asymptotic relationship
involving $\vartheta(t) - \lfloor t \rfloor $ and $t^{1/2}$.
