An explicit upper bound for the first $k$-Ramanujan prime

TL;DR

This paper derives an explicit upper bound for the first k-Ramanujan prime using recent results on prime distribution in small intervals, contributing to the understanding of prime bounds.

Contribution

It provides a new explicit upper bound for the first k-Ramanujan prime based on recent prime interval results.

Findings

Established an explicit upper bound for R_1^{(k)}

Utilized recent prime existence results in small intervals

Enhanced understanding of prime distribution bounds

Abstract

In this paper we establish an explicit upper bound for the first $k$-Ramanujan prime $R_1^{(k)}$ by using a recent result concerning the existence of prime numbers in small intervals.