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Remarks on Ramanujan's inequality concerning the prime counting function

Mehdi Hassani — 2021

Communications in Mathematics

In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π ( x ) 2 < e x log x π x e for x sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor x log x on its right hand side by the factor x log x - h for a given h , and by replacing the numerical factor e by a given positive α . Finally, we introduce and study inequalities analogous...

Some results on derangement polynomials

Mehdi Hassani;  Hossein Moshtagh;  Mohammad Ghorbani — 2022

Commentationes Mathematicae Universitatis Carolinae

We study moments of the difference D n ( x ) - x n n ! e - 1 / x concerning derangement polynomials D n ( x ) . For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for x > 0 . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

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