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.

Key words and phrases. Arithmetic function, summatory function, convolution, square root, Prime Number Theorem.