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

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 $n\gg 1$.