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On a problem of Chowla and some related problems

Published online by Cambridge University Press:  24 October 2008

Paul Erdös
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Let d(m) denote the number of divisors of the integer m. Chowla has conjectured that the integers for which d(m + 1) > d(m) have density ½. In this paper I prove and generalize this conjecture. I prove in § 1 a corresponding result for a general class of functions f(m), and in § 2 the result for d(m) which is not included among the f(m). I employ the method used in my paper: “On the density of some sequences of numbers.”

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 4 , December 1936 , pp. 530 - 540
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

* Journal London Math. Soc. 10 (1935), 120–125.Google Scholar

* The lemma asserts that for every ε we can find a δ such that the number of integers m ≤ n for which c ≤ f (m) ≤ c + δ is less than εn. See also my paper “On the density of some sequences of numbers, II”, which will appear shortly in the Journal of the London Math. Soc.

* More generally we can prove the following theorem. Let X (n) be an arbitrary function with . Then, for almost all integers m ≤ n,

The first inequality may be proved by similar but stronger lemmas than Lemmas 3 and 4. The second inequality has been proved by P. Turán as follows:

which immediately establishes the result.