V. V. Miasoyedov

Hardy-Littlewood inequality for primes

In the article we establish the Hardy-Littlewood inequality $ π(x + y) \leq π(x) + π(y) $. We also prove that the naturally ordered primes $p_1=2,p_2=3,p_3=5,p_4=7,\dots$ satisfy the inequality $ p_ {a + b}> p_a + p_b $ for all $a, \ b \geq 2$.