10.46298/hrj.1988.103
https://hrj.episciences.org/103
Balasubramanian, R
R
Balasubramanian
Ramachandra, K
K
Ramachandra
Some local-convexity theorems for the zeta-function-like analytic functions
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.
episciences.org
functional equation
analytic functions
local-convexity
[MATH] Mathematics [math]
2015-06-12
1988-01-01
1988-01-01
en
journal article
https://hal.archives-ouvertes.fr/hal-01104306v1
2804-7370
https://hrj.episciences.org/103/pdf
VoR
application/pdf
Hardy-Ramanujan Journal
Volume 11 - 1988
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