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Correlations of multiplicative functions and applications

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  31 May 2017

Oleksiy Klurman
Oleksiy Klurman*
Affiliation:
Départment de Mathématiques et de Statistique, Université de Montréal, CP 6128, succ. Centre-ville, Montréal, QC H3C 3J7, Canada email [email protected] Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
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Abstract

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.

Keywords

multiplicative functionsDelange’s theoremcorrelations

MSC classification

Primary: 11L40: Estimates on character sums
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 8 , August 2017 , pp. 1622 - 1657
Copyright
© The Author 2017 

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References

Brüdern, J., Binary additive problems and the circle method, multiplicative sequences and convergent sieves , in Analytic number theory (Cambridge University Press, Cambridge, 2009), 91–132.Google Scholar
Elliott, P. D. T. A., On the correlation of multiplicative functions , Notas Soc. Mat. Chile 11 (1992), 1–11.Google Scholar
Erdős, P., On the distribution function of additive functions , Ann. of Math. (2) 47 (1946), 1–20.Google Scholar
Erdős, P., Some unsolved problems , Michigan Math. J. 4 (1957), 291–300.Google Scholar
Erdős, P., On some of my problems in number theory I would most like to see solved , in Number theory (Ootacamund, 1984), Lecture Notes in Mathematics, vol. 1122 (Springer, Berlin, 1985), 74–84.CrossRefGoogle Scholar
Erdős, P., Some applications of probability methods to number theory , in Mathematical statistics and applications, Vol. B (Bad Tatzmannsdorf, 1983) (Reidel, Dordrecht, 1985), 1–18.Google Scholar
Granville, A. and Soundararajan, K., Large character sums: pretentious characters and the Pólya–Vinogradov theorem , J. Amer. Math. Soc. 20 (2007), 357–384.CrossRefGoogle Scholar
Granville, A. and Soundararajan, K., Sieving and the Erdős–Kac theorem , in Equidistribution in number theory, an introduction, NATO Science Series II: Mathematics, Physics and Chemistry, vol. 237 (Springer, Dordrecht, 2007), 15–27.Google Scholar
Halász, G., On the distribution of additive and the mean values of multiplicative arithmetic functions , Studia Sci. Math. Hungar. 6 (1971), 211–233.Google Scholar
Halász, G., On the distribution of additive arithmetic functions , Acta Arith. 27 (1975), 143–152; collection of articles in memory of Juriĭ Vladimirovič Linnik.CrossRefGoogle Scholar
Hildebrand, A., An Erdős–Wintner theorem for differences of additive functions , Trans. Amer. Math. Soc. 310 (1988), 257–276.Google Scholar
Hildebrand, A., Multiplicative functions at consecutive integers. II , Math. Proc. Cambridge Philos. Soc. 103 (1988), 389–398.Google Scholar
Kátai, I., On a problem of P. Erdős , J. Number Theory 2 (1970), 1–6.Google Scholar
Kátai, I., Some problems in number theory , Studia Sci. Math. Hungar. 16 (1983), 289–295.Google Scholar
Kátai, I., Multiplicative functions with regularity properties. VI , Acta Math. Hungar. 58 (1991), 343–350.Google Scholar
Kátai, I., Continuous homomorphisms as arithmetical functions, and sets of uniqueness , in Number theory, Trends in Mathematics (Birkhäuser, Basel, 2000), 183–200.Google Scholar
Matomäki, K. and Radziwiłł, M., Multiplicative functions in short intervals , Ann. of Math. (2) 183 (2016), 1015–1056.Google Scholar
Matomäki, K., Radziwiłł, M. and Tao, T., An averaged form of Chowla’s conjecture , Algebra Number Theory 9 (2015), 2167–2196.Google Scholar
Mauclaire, J.-L. and Murata, L., On the regularity of arithmetic multiplicative functions. I , Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 438–440.Google Scholar
Phong, B., A characterization of some unimodular multiplicative functions , Publ. Math. Debrecen 57 (2000), 339–366.Google Scholar
Phong, B. M., Additive functions at consecutive integers , Acta Math. Hungar. 142 (2014), 260–274.CrossRefGoogle Scholar
Stepanauskas, G., The mean values of multiplicative functions. V , in Analytic and probabilistic methods in number theory (Palanga, 2001) (TEV, Vilnius, 2002), 272–281.Google Scholar
Tao, T., The Erdős discrepancy problem , Discrete Anal. 1 (2016), 27pp, arXiv:1509.05363.Google Scholar
Tao, T., The logarithmically averaged Chowla and Elliott conjectures for two-point correlations , Forum Math. Pi 4 (2016), 36pp.CrossRefGoogle Scholar
Wirsing, E., Tang, Y. and Shao, P., On a conjecture of Kátai for additive functions , J. Number Theory 56 (1996), 391–395.CrossRefGoogle Scholar
Wirsing, E. and Zagier, D., Multiplicative functions with difference tending to zero , Acta Arith. 100 (2001), 75–78.Google Scholar