Hardy-Littlewood inequality for primes

TL;DR

This paper proves a Hardy-Littlewood inequality relating the prime counting function and demonstrates a new inequality involving the ordered primes, showing that the sum of primes exceeds the prime at the sum of their indices.

Contribution

It establishes a novel inequality for the prime counting function and a new inequality for ordered primes, expanding understanding of prime distribution.

Findings

Proves $\pi(x + y) \leq \pi(x) + \pi(y)$ for the prime counting function.

Shows that for all $a, b \geq 2$, $p_{a + b} > p_a + p_b$ holds for ordered primes.

Provides new inequalities that relate prime indices and values.

Abstract

In the article we establish the Hardy-Littlewood inequality $ \pi (x + y) \leq \pi (x) + \pi (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 $a, \ b \geq 2$.