An inequality for the function π(n)

Abstract

We prove that the inequality $\pi ^2 \left( m \right) + \pi ^2 \left( n \right) \leqslant \tfrac{5} {4}\pi ^2 \left( {m + n} \right)$ holds for all integers m, n ≥ 2. The constant factor 5/4 is sharp. This complements a result of Panaitopol, who showed in 2001 that ½ π ²(m+ n) ≤ π ²(m) + π ²(n) is valid for all m, n ≥ 2. Here, as usual, π(n) denotes the number of primes not exceeding n.