Correlations of multiplicative functions and applications
By Oleksiy Klurman. Compositio Mathematica, 2016
55 citations recorded by the source corpus.
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 $\operatorname{Re}(s)<1$ 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.
Subfields: Algebra and Number Theory, Applied Mathematics, Management Science and Operations Research
View Paper: https://www.semanticscholar.org/paper/95154c26d8723d03955db0b1da8eab4037e9606d
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In Algebra and Number Theory
- Joni Teravainen
- Terence Tao
- Nikos Frantzikinakis
- B. Host
- Ben Joseph Green
- T. Ziegler
- Dorin Ervin Dutkay
- William T. Gowers
In Applied Mathematics
- Joni Teravainen
- Terence Tao
- Sanghyuk Lee
- D. Tataru
- Zihua Guo
- Carlos E. Kenig
- Jonathan Bennett
- Shuanglin Shao
In Management Science and Operations Research
Citations in the corpus
- The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures (2017, background): “erages over independent polynomials in several variables in [11]. The papers [10], [29] in turn established two-dimensional variants of Elliott’s conjecture. We also mentionthe recent paper of Klurman[24] whichgivesan explicitformula for the limit (1) in the “pretentious” case that (2) holds for all”