Rounding the arithmetic mean value of the square roots of the first $n$ integers

TL;DR

This paper investigates the average of the square roots of the first n integers, providing an asymptotic formula, a conjectured integer part formula, and methods for numerical evaluation for large n.

Contribution

It introduces a new asymptotic expression for the mean of square roots and confirms a recent conjecture about its integer part, enhancing understanding of this mathematical quantity.

Findings

Derived an asymptotic expression for the mean value

Proved a formula for the integer part of the mean

Developed methods for efficient numerical evaluation for large n

Abstract

In this article we study the arithmetic mean value $\Sigma(n)$ of the square roots of the first $n$ integers. For this quantity, we develop an asymptotic expression, and derive a formula for its integer part which has been conjectured recently in the work of M. Merca. Furthermore, we address the numerical evaluation of $\Sigma(n)$ for large $n\gg 1$.