Value patterns of multiplicative functions and related sequences


Tao, T
Teräväinen, J

Publication Date: 

26 September 2019


Forum of Mathematics, Sigma

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We study the existence of various sign and value patterns in sequences
defined by multiplicative functions or related objects. For any set $A$ whose
indicator function is 'approximately multiplicative' and uniformly distributed
on short intervals in a suitable sense, we show that the asymptotic density of
the pattern $n+1\in A$, $n+2\in A$, $n+3\in A$ is positive, as long as $A$ has
density greater than $\frac{1}{3}$. Using an inverse theorem for sumsets and
some tools from ergodic theory, we also provide a theorem that deals with the
critical case of $A$ having density exactly $\frac{1}{3}$, below which one
would need nontrivial information on the local distribution of $A$ in Bohr sets
to proceed. We apply our results firstly to answer in a stronger form a
question of Erd\H{o}s and Pomerance on the relative orderings of the largest
prime factors $P^{+}(n)$, $P^{+}(n+1), P^{+}(n+2)$ of three consecutive
integers. Secondly, we show that the tuple
$(\omega(n+1),\omega(n+2),\omega(n+3)) \pmod 3$ takes all the $27$ possible
patterns in $(\mathbb{Z}/3\mathbb{Z})^3$ with positive lower density, with
$\omega(n)$ being the number of distinct prime divisors. We also prove a
theorem concerning longer patterns $n+i\in A_i$, $i=1,\dots k$ in approximately
multiplicative sets $A_i$ having large enough densities, generalising some
results of Hildebrand on his 'stable sets conjecture'. Lastly, we consider the
sign patterns of the Liouville function $\lambda$ and show that there are at
least $24$ patterns of length $5$ that occur with positive density. In all of
the proofs we make extensive use of recent ideas concerning correlations of
multiplicative functions.

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Journal Article