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Pretentiously detecting power cancellation

Published online by Cambridge University Press:  17 January 2013

JUNEHYUK JUNG  and
ROBERT J. LEMKE OLIVER
JUNEHYUK JUNG
Affiliation:
Princeton University, Department of Mathematics, Princeton, NJ 08544, U.S.A. e-mail: junehyuk@math.princeton.edu
ROBERT J. LEMKE OLIVER
Affiliation:
Emory University, Department of Mathematics and Computer Science, Atlanta, GA 30322, U.S.A. e-mail: rlemkeo@emory.edu
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Abstract

Granville and Soundararajan have recently introduced the notion of pretentiousness in the study of multiplicative functions of modulus bounded by 1, essentially the idea that two functions which are similar in a precise sense should exhibit similar behavior. It turns out, somewhat surprisingly, that this does not directly extend to detecting power cancellation - there are multiplicative functions which exhibit as much cancellation as possible in their partial sums that, modified slightly, give rise to functions which exhibit almost as little as possible. We develop two new notions of pretentiousness under which power cancellation can be detected, one of which applies to a much broader class of multiplicative functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 3 , May 2013 , pp. 481 - 498
Copyright
Copyright © Cambridge Philosophical Society 2013

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References

REFERENCES

[1]Granville, A.Pretentiousness in analytic number theory. J. Théor. Nombres Bordeaux 21 (1) (2009), 159–173.CrossRefGoogle Scholar
[2]Granville, A. and Soundararajan, K.Multiplicative number theory. in preparation.Google Scholar
[3]Granville, A. and Soundararajan, K.The spectrum of multiplicative functions. Ann. of Math. (2) 153 (2) (2001), 407–470.CrossRefGoogle Scholar
[4]Granville, A. and Soundararajan, K.Decay of mean values of multiplicative functions. Canad. J. Math. 55 (6) 2003, 1191–1230.CrossRefGoogle Scholar
[5]Granville, A. and Soundararajan, K.Large character sums: pretentious characters and the Pólya-Vinogradov theorem. J. Amer. Math. Soc. 20 (2) (electronic) (2007), 357–384.CrossRefGoogle Scholar
[6]Granville, A. and Soundararajan, K.Pretentious multiplicative functions and an inequality for the zeta-function. In Anatomy of Integers, CRM Proc. Lecture Notes, vol. 46 (Amer. Math. Soc., Providence, RI, 2008), pages 191–197.Google Scholar
[7]Halász, G.Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403.CrossRefGoogle Scholar
[8]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
[9]Koukoulopoulos, D. On multiplicative functions which are small on average. Preprint.Google Scholar
[10]Montgomery, H.A note on mean values of multiplicative functions. Report No. $17$, Institut Mittag-Leffler, Djursholm (1978).Google Scholar
[11]Montgomery, H. L. and Vaughan, R. C.Multiplicative Number Theory. I. Classical Theory. Cambridge Studies in Advanced Math. vol 97 (Cambridge University Press, Cambridge, 2007)Google Scholar
[12]Tenenbaum, G.Introduction to Analytic and Probabilistic Number Theory, vol 46 Cambridge Studies in Advanced Math. (Cambridge University Press, Cambridge, 1995).Google Scholar