  We prove a lower bound on the supremum of the function $S_1(T)$ on short
intervals, defined by the integration of the argument of the Riemann
zeta-function. The same type of result on the supremum of $S(T)$ have already
been obtained by Karatsuba and Korolev. Our result is based on the idea of the
paper of Karatsuba and Korolev. Also, we show an improved Omega-result for a
lower bound.
