  We combine our version of the resonance method with certain convolution
formulas for $\zeta(s)$ and $\log\, \zeta(s)$. This leads to a new $\Omega$
result for $|\zeta(1/2+it)|$: The maximum of $|\zeta(1/2+it)|$ on the interval
$1 \le t \le T$ is at least $\exp\left((1+o(1)) \sqrt{\log T \log\log\log
T/\log\log T}\right)$. We also obtain conditional results for $S(t):=1/\pi$
times the argument of $\zeta(1/2+it)$ and $S_1(t):=\int_0^t S(\tau)d\tau$. On
the Riemann hypothesis, the maximum of $|S(t)|$ is at least $c \sqrt{\log T
\log\log\log T/\log\log T}$ and the maximum of $S_1(t)$ is at least $c_1
\sqrt{\log T \log\log\log T/(\log\log T)^3}$ on the interval $T^{\beta} \le t
\le T$ whenever $0\le \beta < 1$.
