On a certain asymptotic relationship involving $\vartheta(t) - \lfloor t   \rfloor$ and $t^{1/2}$
  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 $t^{1/2}$. 
Hisanobu Shinya