  The L\'evy hypothesis states that inverse square L\'evy walks are optimal
search strategies because they maximise the encounter rate with sparse,
randomly distributed, replenishable targets. It has served as a theoretical
basis to interpret a wealth of experimental data at various scales, from
molecular motors to animals looking for resources, putting forward the
conclusion that many living organisms perform L\'evy walks to explore space
because of their optimal efficiency. Here we provide analytically the
dependence on target density of the encounter rate of L\'evy walks for any
space dimension $d$ ; in particular, this scaling is shown to be {\it
independent} of the L\'evy exponent $\alpha$ for the biologically relevant case
$d\ge 2$, which proves that the founding result of the L\'evy hypothesis is
incorrect. As a consequence, we show that optimizing the encounter rate with
respect to $\alpha$ is {\it irrelevant} : it does not change the scaling with
density and can lead virtually to {\it any} optimal value of $\alpha$ depending
on system dependent modeling choices. The conclusion that observed inverse
square L\'evy patterns are the result of a common selection process based
purely on the kinetics of the search behaviour is therefore unfounded.
