  We show that the $L^1$ norm of an exponential sum of length $X$ and with
coefficients equal to the Liouville or M\"{o}bius function is at least
$\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$.
  For the Liouville function this improves on the lower bound $\gg
X^{c/\log\log X}$ due to Balog and Perelli (1998). For the M\"{o}bius function
this improves the lower bound $\gg X^{1/6}$ due to Balog and Ruzsa (2001). The
large discrepancy between these lower bounds is due to the method employed by
Balog and Ruzsa, as it crucially relies on the vanishing of $\mu(n)$. Instead
our proof puts the two cases on an equal footing by exploiting the connection
of these coefficients with zeros of Dirichlet $L$-functions. In the second
paper in this series we will obtain a lower bound $\gg X^{\delta}$ for some
small $\delta$ but for general (non-pretentious) multiplicative functions.
