A footnote to a theorem of Hal\'{a}sz

We study multiplicative functions $f$ satisfying $\mathrm{\mid }f(n)\mathrm{\mid }\le 1$ for all $n$, the associated Dirichlet series $F(s):={\sum }_{n=1}^{\mathrm{\infty }}f(n)n^{-s}$, and the summatory function $S_f(x):={\sum }_{n\le x}f(n)$. Up to a possible trivial contribution from the numbers $f(2^k)$, $F(s)$ may have at most one zero or one pole on the one-line, in a sense made precise by Hal'{a}sz. We estimate $\log F(s)$awayfromanysuchpointandshowthatif$F(s)$ has a zero on the one-line in the sense of Hal'{a}sz, then $|S_f(x)|\le (x/\log x) \exp\big(c\sqrt{\log \log x}\big)$forall$c>0$when$x$ is large enough. This bound is best possible.