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Keywords:
Prime numbers; Ramanujan's inequality; Riemann hypothesis
Prime numbers; Ramanujan's inequality; Riemann hypothesis
Summary:
In this paper we investigate Ramanujan's inequality concerning the prime counting function, asserting that $\pi (x)^2<\frac {\mathrm{e} \,x}{\log x}\,\pi \left (\frac {x}{\mathrm{e} }\right )$ 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 $\frac {x}{\log x}$ on its right hand side by the factor $\frac {x}{\log x-h}$ for a given $h$, and by replacing the numerical factor $\mathrm{e} $ by a given positive $\alpha $. Finally, we introduce and study inequalities analogous to Ramanujan's inequality.
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