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Divisor function inequalities, entropy, and the chance of being below average

Published online by Cambridge University Press:  01 March 2017

ZARATHUSTRA BRADY
ZARATHUSTRA BRADY*
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Bldg. 380, Stanford, CA 94305-2125, U.S.A. e-mail: [email protected]
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Abstract

We extend a lower bound of Munshi on sums over divisors of a number n which are less than a fixed power of n from the squarefree case to the general case. In the process we prove a lower bound on the entropy of a geometric distribution with finite support, as well as a lower bound on the probability that a random variable is less than its mean given that it satisfies a natural condition related to its third cumulant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 547 - 560
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Alon, N., Huang, H. and Sudakov, B.. Nonnegative k-sums, fractional covers, and probability of small deviations. J. Combin. Theory Ser. B 102 3 (2012), 784–796.Google Scholar
[2] FEDJA (http://mathoverflow.net/users/1131/fedja). lower-bound for Pr[X ≥ EX]. MathOverflow. URL:http://mathoverflow.net/q/188087 (version: 2014-11-26).Google Scholar
[3] Friedlander, J. B. and Iwaniec, H.. Divisor weighted sums. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 322. Trudy po Teorii Chisel (2005), 212–219, 255.Google Scholar
[4] Hartke, S. G. and Stolee, D.. A linear programming approach to the Manickam–Miklós–Singhi conjecture. European J. Combin. 36 (2014), 53–70.Google Scholar
[5] Khovanskiĭ, A. G.. Fewnomials. Trans. Math. Monogr. vol. 88 (American Mathematical Society, Providence, RI, 1991). Translated from the Russian by Smilka Zdravkovska.Google Scholar
[6] Landreau, B.. A new proof of a theorem of van der Corput. Bull. London Math. Soc. 21 4 (1989), 366–368.Google Scholar
[7] Munshi, R.. Inequalities for divisor functions. Ramanujan J. 25 2 (2011), 195–201.Google Scholar
[8] Pokrovskiy, A.. A linear bound on the Manickam–Miklos–Singhi Conjecture. ArXiv e-prints (Aug. 2013).Google Scholar
[9] Soundararajan, K.. An inequality for multiplicative functions. J. Number Theory 41 2 (1992), 225–230.Google Scholar
[10] Widder, D. V.. The Laplace Transform. Princeton Mathematical Series, v. 6 (Princeton University Press, Princeton, N. J., 1941).Google Scholar
[11] Wolke, D.. A new proof of a theorem of van der Corput. J. London Math. Soc. s2-5 4 (1972), 609–612.Google Scholar