Home
Hostname: page-component-79b67bcb76-c5xhk Total loading time: 0.254 Render date: 2021-05-15T23:49:50.397Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

# Correlations of multiplicative functions and applications

Published online by Cambridge University Press:  31 May 2017

Corresponding

## 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.

## MSC classification

Type
Research Article
Information
Compositio Mathematica , August 2017 , pp. 1622 - 1657
© The Author 2017

## Access options

Get access to the full version of this content by using one of the access options below.

## References

Brüdern, J., Binary additive problems and the circle method, multiplicative sequences and convergent sieves , in Analytic number theory (Cambridge University Press, Cambridge, 2009), 91132.Google Scholar
Elliott, P. D. T. A., On the correlation of multiplicative functions , Notas Soc. Mat. Chile 11 (1992), 111.Google Scholar
Erdős, P., On the distribution function of additive functions , Ann. of Math. (2) 47 (1946), 120.CrossRefGoogle Scholar
Erdős, P., Some unsolved problems , Michigan Math. J. 4 (1957), 291300.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), 7484.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), 118.Google Scholar
Granville, A. and Soundararajan, K., Large character sums: pretentious characters and the Pólya–Vinogradov theorem , J. Amer. Math. Soc. 20 (2007), 357384.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), 1527.Google Scholar
Halász, G., On the distribution of additive and the mean values of multiplicative arithmetic functions , Studia Sci. Math. Hungar. 6 (1971), 211233.Google Scholar
Halász, G., On the distribution of additive arithmetic functions , Acta Arith. 27 (1975), 143152; 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), 257276.Google Scholar
Hildebrand, A., Multiplicative functions at consecutive integers. II , Math. Proc. Cambridge Philos. Soc. 103 (1988), 389398.CrossRefGoogle Scholar
Kátai, I., On a problem of P. Erdős , J. Number Theory 2 (1970), 16.CrossRefGoogle Scholar
Kátai, I., Some problems in number theory , Studia Sci. Math. Hungar. 16 (1983), 289295.Google Scholar
Kátai, I., Multiplicative functions with regularity properties. VI , Acta Math. Hungar. 58 (1991), 343350.CrossRefGoogle Scholar
Kátai, I., Continuous homomorphisms as arithmetical functions, and sets of uniqueness , in Number theory, Trends in Mathematics (Birkhäuser, Basel, 2000), 183200.Google Scholar
Matomäki, K. and Radziwiłł, M., Multiplicative functions in short intervals , Ann. of Math. (2) 183 (2016), 10151056.CrossRefGoogle Scholar
Matomäki, K., Radziwiłł, M. and Tao, T., An averaged form of Chowla’s conjecture , Algebra Number Theory 9 (2015), 21672196.CrossRefGoogle Scholar
Mauclaire, J.-L. and Murata, L., On the regularity of arithmetic multiplicative functions. I , Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 438440.CrossRefGoogle Scholar
Phong, B., A characterization of some unimodular multiplicative functions , Publ. Math. Debrecen 57 (2000), 339366.Google Scholar
Phong, B. M., Additive functions at consecutive integers , Acta Math. Hungar. 142 (2014), 260274.CrossRefGoogle Scholar
Stepanauskas, G., The mean values of multiplicative functions. V , in Analytic and probabilistic methods in number theory (Palanga, 2001) (TEV, Vilnius, 2002), 272281.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), 391395.CrossRefGoogle Scholar
Wirsing, E. and Zagier, D., Multiplicative functions with difference tending to zero , Acta Arith. 100 (2001), 7578.CrossRefGoogle Scholar

# Send article to Kindle

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Correlations of multiplicative functions and applications
Available formats
×

# Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Correlations of multiplicative functions and applications
Available formats
×

# Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Correlations of multiplicative functions and applications
Available formats
×
×