Correlations of multiplicative functions and applications
We give an asymptotic formula for correlations $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$ where $f,\ldots ,f_{m}$ are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several des...
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Veröffentlicht in: | Compositio mathematica 2017-08, Vol.153 (8), p.1622-1657 |
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Format: | Artikel |
Sprache: | eng |
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Online-Zugang: | Volltext |
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Zusammenfassung: | We give an asymptotic formula for correlations
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$
where
$f,\ldots ,f_{m}$
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) |
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ISSN: | 0010-437X 1570-5846 |
DOI: | 10.1112/S0010437X17007163 |