New bounds for the sum of the first $n$ prime numbers

TL;DR

This paper derives a new asymptotic formula for the sum of the first n primes, generalizes existing formulas, and improves explicit bounds using results related to Mandl's inequality.

Contribution

It introduces a generalized asymptotic formula for prime sums and provides improved explicit estimates, advancing the understanding of prime number summations.

Findings

New asymptotic formula for sum of first n primes

Generalization of Massias and Robin's formula

Enhanced explicit bounds for prime sums

Abstract

In this paper we establish a general asymptotic formula for the sum of the first $n$ prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin. Further we prove a series of results concerning Mandl's inequality on the sum of the first $n$ prime numbers. We use these results to find new explicit estimates for the sum of the first $n$ prime numbers, which improve the currently best known estimates.