Let
denote
the Liouville function. A well-known conjecture of Chowla asserts that, for any distinct natural
numbers
,
one has
as
. This conjecture
remains unproven for any
with
.
Using the recent results of Matomäki and Radziwiłł on mean values of
multiplicative functions in short intervals, combined with an argument of Kátai and
Bourgain, Sarnak, and Ziegler, we establish an averaged version of this conjecture,
namely
as
, whenever
goes to
infinity as
and
is
fixed. Related to this, we give the exponential sum estimate
as
uniformly
for all
,
with
as
before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the
order of
)
and extend to more general bounded multiplicative functions than the Liouville
function, yielding an averaged form of a (corrected) conjecture of Elliott.
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