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A good day to everyone!

I have a (somewhat) intriguing question regarding Paul Erdős's papers on Egyptian fractions (e.g., his 2nd paper during his mathematical career was about this topic).

My question is: What was his (primary) motivation for pursuing this research topic?

In case no historian nor book on the history of mathematics can provide us a definite answer, will it be safe to assume that Erdős was trying to build up a whole theory (and comprehensive framework) to solve the Odd Perfect Number (OPN) conjecture, given the trend in his succeeding research topics in elementary number theory?

I quote Melvin Nathanson who, in his book titled "Elementary Methods in Number Theory", described Erdős as being a "master of elementary methods". In particular, Erdős was able to come up with a proof of the Prime Number Theorem using a purely elementary approach - although, this did not equate to a comparatively simple and/or easy proof.

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What paper are you talking about? I can't find any of his papers about Egyptian fractions until the 70s. – Charles Mar 5 '13 at 14:29
Erdos was interested in an extremely wide range of topics. If he left no hint, answering this is probably hopeless. In any case, he has been described as a "problem solver" in contrast to "theory builders", so I'd guess he wasn't after any wide theory here. But as they say, your guess is as good as mine. – vonbrand Mar 5 '13 at 14:35
@Charles, you can try to refer to the following paper by Ron Graham, who was a collaborator of Paul Erdos. The paper I allude to was published in 1932. The paper is in Hungarian, and is titled "Generalization of an elementary number-theoretic theorem of Kurschak". You might want to check for a copy of this paper in the Erdos Project. =) – user11235813 Mar 5 '13 at 15:32
@vonbrand, if he was indeed a "problem solver" rather than a "theory builder", then I am (of course) very surprised (today) that most of the results that I need to settle certain "cases" of the OPN Conjecture are either contained in one of his papers, joint articles with his collaborators, or in the latest and not-so-recent works of some of his collaborators, as of 2013. – user11235813 Mar 5 '13 at 15:38
@ArnieB.Dris, that means that he solved many problems in that general area, not that he left any sort of comprehensive theory behind. – vonbrand Mar 5 '13 at 15:51
up vote 4 down vote accepted

Prior to the full proof of the PNT, the best estimates for $\pi(n)$ came from Chebyshev-type bounds. Chebyshev (1852) published the bounds

$$0.921292 \frac{x}{\log x} \ll \pi(x) \ll 1.10555\frac{x}{\log x},$$

as $x \to \infty$, and the constants given here were later improved by Sylvester. Then 1892 came, and Chebyshev's techniques were for the most part forgotten.

Around 1937, Erdős showed that Chebyshev-type methods had the potential to prove the PNT, in that the constants above could be taken arbitrarily close to $1$ in the line above. This came with a caveat, though, in that Erdős had assumed the PNT to reach this result (so it was, for the most part, an academic exercise).

In particular, Erdős used an equivalent form of the PNT - that the function

$$ M(n):=\sum_{k \leq n} \mu(n)$$ is $o(n)$. To see where this might come up, recall that Chebyshev obtained the above bounds by consideration of the function

$$ \chi(t):= \psi(t)-\frac{1}{2} \psi(t)-\frac{1}{3}\psi(t)-\frac{1}{5}\psi(t)+\frac{1}{30}\psi(t).$$

Regarding the coefficients, we note that

  1. They sum to $1$. This is necessary to obtain an estimate of the appropriate growth order.
  2. They fit the form $1/\mu(k)$, for $k \leq 5$. This relates to why Chebyshev's bounds are so good; in particular, this explains why the ratio of the upper and lower bounds given above is $6/5$.

As for the term $1/30$, this is only used to satisfy (1). Erdős tweaked Chebyshev's idea by considering variants of the $\chi$ function that have general rational coefficients, and it is around here that the growth of the $\mu$ function must be considered.

Disclaimer - Part of the following (the non-historical bits) is conjecture. All the same, it represents my experience with Erdős's work on the elementary PNT:

The set $S=\{1,-2,-3,-5\}$ is very special in that the harmonic sum of $S$ is the reciprocal of an integer. In general, the set $$S_k=\{1,\mu(2),\mu(3),\ldots,\mu(k)\}$$ will not have this property, and we can ask just how many non-zero integers must be added to $S_k$ such that the resulting set has $0$ harmonic sum. (To avoid trivializing this question, we should assume that the new integers $\{n_i\}$ satisfy $\vert n_i \vert >k$.)

If this problem had a nice solution, Erdős would have likely found a novel proof of the PNT. Unfortunately, this problem is difficult, and it is difficult precisely because Egyptian fractions are still poorly understood.

This obstruction could have easily suggested to Erdős the importance of Egyptian fractions in number theory.

Here are some places you may wish to pursue for further reading:

  1. P. L. Chebyshev, Mémoire sur les Nombres Premiers, J. Math. Pures Appl., 17, (1852). (This is Chebyshev's original paper on the topic.)

  2. H. Diamond and P. Erdős, On Sharp Elementary Prime Number Estimates, L’Enseignement Mathématique, 26, (1980). (This includes Erdős's work on the Chebyshev method, in relation to previously established proofs of the PNT.)

  3. A recent blog post of mine, in which Chebyshev's method is generalized to the point at which Egyptian fractions enter as a limiting factor. This article also hashes out many of the details glossed over in this post.

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My profuse thanks @A Walker for your detailed answer and the references! I am hereby accepting your answer... :) – user11235813 Mar 6 '13 at 18:22
Just a comment on what appears to be an error that I just recently spotted @AWalker, shouldn't the indicated year in the second sentence, first paragraph of your answer be 1892 instead of 1982? I apologize, I do realize it may have been a typo from your end... – user11235813 Mar 31 '13 at 5:04
@ArnieB.Dris You're completely correct. I'll note the edit above. The PNT was proven (twice, independently) in 1892. – A Walker Apr 1 '13 at 6:50

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