Some new inequalities for primes

TL;DR

This paper introduces new inequalities involving prime numbers, providing bounds that relate the n-th prime to sums of previous primes, enhancing understanding of prime distribution.

Contribution

The paper presents novel inequalities for prime numbers, establishing bounds that improve existing knowledge about prime distributions for large n.

Findings

p_n > n + sum_{k=1}^n p_k / k for all n > 124

sum_{k=1}^n k p_k < n^2 p_n / 3 for all n > 30

Provides bounds that could influence future prime number research

Abstract

For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all n>30.