The author investigates the behaviour of the \(L^ 1\)-mean \(\int^{1}_{0}| S(x)| dx\) of the exponential sum \(S(x) = \sum_{n\leq X}\Lambda(n)e^{2\pi inx}\), where \(\Lambda\) is von Mangoldt's function. Such means often arise in investigations in analytic number theory. Making use of Cauchy's inequality, Parseval's identity and the prime number theorem, it is easy to verify that \[ \int^{1}_{0}| S(x)| dx\leq (1+o(1))\cdot (X \log X)^{1/2}. \] The purpose of the present paper is to give a simple proof of the following result: There is a positive number \(C\) such that for all \(X\geq 2\) we have \[ \int^{1}_{0}| S(x)| dx\geq C\cdot X^{1/2}. \] Lower bounds for the \(L^ 1\) means of \(\sum_{p\leq X}e^{2\pi ipx}\) and \(\sum_{p\leq X}(\log p)e^{2\pi ipx}\) follow easily from this estimate.