1 results
Correlations of multiplicative functions and applications
- Part of
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 8 / August 2017
- Published online by Cambridge University Press:
- 31 May 2017, pp. 1622-1657
- Print publication:
- August 2017
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- Article
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We give an asymptotic formula for correlations
where
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$
$f,\ldots ,f_{m}$ are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions 
$f:\mathbb{N}\rightarrow \{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from 
$1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of the multiplicative function is zero, then either 
$f(n)=n^{s}$ for 
$\operatorname{Re}(s)<1$ or 
$|f(n)|$ is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of 
$n=a+b$, where 
$a,b$ belong to some multiplicative subsets of 
$\mathbb{N}$. This gives a new ‘circle method-free’ proof of a result of Brüdern.
