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Some local-convexity theorems for the zeta-function-like analytic functions

Abstract : 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.
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R Balasubramanian, K Ramachandra. Some local-convexity theorems for the zeta-function-like analytic functions. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1988, Volume 11 - 1988, pp.1 - 12. ⟨10.1090/conm/210/02800⟩. ⟨hal-01104306⟩

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