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In this paper, we proved an exact asymptotically sharp upper bound of the$L^p$ Lebesgue Constant (i.e. the $L^p$ norm of Dirichlet kernel) for $p\ge 2$.As an application, we also verified the implication of a new $\infty $-R\'enyientropy power inequality for integer valued random variables.

An Exact Upper Bound on the $L^p$ Lebesgue Constant and The $\infty$-R\'enyi Entropy Power Inequality for Integer Valued Random Variables