There is a well-known relationship between the distribution of primes and
distribution of ±1 signs of the Liouville function (the
completely multiplicative function taking the value -1 at all primes).
A conjecture of Chowla, analogising the Hardy-Littlewood
prime k-tuples conjecture, predicts that
the autocorrelations of λ, e.g.
1⁄x ∑n ≤ x
λn+1 ... λn+k tend
to 0 on average as x tends to ∞.
This conjecture, along with its generalisation to the broader collection of
bounded "non-pretentious" multiplicative functions, due
originally to Elliott, remain wide open for k ≥ 2. Previously,
there were no explicit examples in the literature of (deterministic and
scale-independent) non-pretentious multiplicative functions known to
satisfy Elliott's conjecture.
In this talk I will present a const-
ruction of a non-pretentious multiplicative function f : N
→ {-1,1} all of whose auto-correlations tend to 0 on average, answering
a(n ergodic theory) question of Lemanczyk and de la Rue. I will further discuss
the following applications of this construction:
i) a proof that Chowla's conjecture does not imply the
Riemann Hypothesis, i.e., there are ±1-valued multiplicative
functions f all of
whose autocorrelations tend to 0, but that do not exhibit square-root
cancellation on average (the object of some recent speculation);
ii) there are multiplicative subsemigroups of N with Poissonian gap
statistics, thus giving an unconditional multiplicative analogue
of a classical result of Gallagher about primes in short intervals.
(Joint work with Oleksiy Klurman, Par Kurlberg and Joni Teravainen.)