Convolution square root of \(1\) and the
Prime Number Theorem

Abstract. Define \(1^{*1/2}\), the convolution square root of \(1\), as the arithmetic function satisfying \(1^{*1/2} * 1^{*1/2} = 1\) with \(1^{*1/2}(1) = 1\). We discuss properties of this function and show how its summatory function is connected with the Prime Number Theorem.