Geodesic
Inequalities involving π(x)
Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 237-251 Cet article a éte moissonné depuis la source Numdam

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We present several inequalities involving the prime-counting function π(x). Here, we give two examples of our results. We show that

16 9 π ( x ) π ( y ) ≤ π 2 ( x + y )

is valid for all integers x,y≥2. The constant factor 16/9 is the best possible. The special case x=y leads to

4 3 ≤ π ( 2 x ) π ( x ) ( x = 2 , 3 , ... ) ,

where the lower bound 4/3 is sharp. This complements Landau’s well-known inequality

π ( 2 x ) π ( x ) ≤ 2 ( x = 2 , 3 , ... ) .

Moreover, we prove that the inequality

2 π ( x + y ) x + y s ≤ π ( x ) x s + π ( y ) y s ( 0 < s ∈ ℝ )

holds for all integers x,y≥2 if and only if s≤s 0 =0.94745.... Here, s 0 is the only positive solution of

16 7 t - 6 5 t = 1 .
Accepté le :
Publié le :
DOI : 10.4171/rsmup/98
Classification : 11 
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     author = {Alzer, Horst and Kwong, Man Kam and S\'andor, J\'ozsef},
     title = {Inequalities involving $\pi(x)$
            },
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {237--251},
     year = {2022},
     volume = {147},
     doi = {10.4171/rsmup/98},
     mrnumber = {4450790},
     zbl = {1497.11014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/rsmup/98/}
}
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Alzer, Horst; Kwong, Man Kam; Sándor, József. Inequalities involving $\pi(x)$. Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 237-251. doi: 10.4171/rsmup/98

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