In this paper we investigate lower bounds for $$I(\sigma)= \int^H_{-H}\vert f(\sigma+it_0+iv)\vert^kdv,$$ where $f(s)$ is analytic for $s=\sigma+it$ in $\mathcal{R}=\{a\leq\sigma\leq b, t_0-H\leq t\leq t_0+H\}$ with $\vert f(s)\vert\leq M$ for $s\in\mathcal{R}$. Our method rests on a convexity technique, involving averaging with the exponential function. We prove a general lower bound result for $I(\sigma)$ and give an application concerning the Riemann zeta-function $\zeta(s)$. We also use our methods to prove that large values of $\vert\zeta(s)\vert$ are ``rare'' in a certain sense.

Large values estimates of $|\zeta(\sigma + it)|$ over a set of well-spaced points are obtained, when $\sigma$ is close to the line $\sigma = 1$. Analogous results are obtained for zeta-functions of cusp forms and the Dedekind zeta-funciton.

Let $[a_1(0),\ldots,a_n(0)]$ be a real vector; define recursively $a_i(t+1)=|a_i(t)-a_{(i+1)}(t)|$. This paper is devoted to the study of $[a_1(t),\ldots,a_n(t)]$ for $t$ tending to infinity.