Curious Constants (part 4 of 4)

I hope you have enjoyed at least parts of this series (links at the bottom). I learned quite a bit along the way, and would like to point you to the following…

Resources

As evidenced by the links above, Wikipedia played a huge role in my preparation for this talk, as did Mathworld. The constant I was originally inspired to talk about was the last one in the talk above, Khinchin’s. I first learned about it in the book “Gamma” by Havil, a book I definitely recommend. This book also introduced me to the first integral ($1/x^x$), and the Kempner series. I have already mentioned Khinchin’s book “Continued Fractions,” which I had quite a great time reading. Once I decided to give a talk on scattered constants, the book “Mathematical Constants” by Finch proved hugely valuable. In addition to bringing many more constants to my attention, each of it’s sections has a huge list of references.

The proof above about the divergence of the series of prime reciprocals was from W. Dunham’s book, “Euler: The Master of Us All,” another great book. The proofs mentioned about the alternating sum of factorials came from the paper “Euler Subdues a Very Obstreperous Series” by E.J. Barbeau, which I found in the book “The Genius of Euler,” (edited by W. Dunham, part of the MAA Tercentenary Euler Celebration). Though I may not have used it explicitly, I regularly peaked into Hardy and Wright’s “An Introduction to the Theory of Numbers” (a book which I now hope to dig into even more). Another fun book, which I consulted but did not use explicitly, is Conway’s “Book of Numbers.”

The results I found the most difficult to grasp were the probability reasonings associated with the twin prime constant and Artin’s conjecture. For Artin’s conjecture, I looked at the paper “Artin’s Conjecture for Primitive Roots”, by M. Ram Murty, available here. As I have already mentioned, the paper “An Amazing Prime Heuristic” by Caldwell (here) was the one that finally made things seem reasonable about the twin prime constant. I also read the papers “The Twin Prime Constant” and “Heuristic Reasoning in the Theory of Numbers”, by Golomb and Polya, respectively. Both of these papers I was able to access through UVA on JSTOR. The book “The Prime Numbers and Their Distribution”, by G. Tenenbaum and M. M. France also had a section on twin primes and also Mertens theorems.

Finally, some journal articles. I apologize for any incorrect formatting.

• P. Erdos, On the density of the abundant numbers, J. London Math. Soc. 9 (1934), 278-282.
• G.N. Watson, Theorems stated by Ramanujan. VIII: Theorems on divergent series, J. London Math. Soc. 4 (1929) 82-86.
• M. Schroeder, How Probable is Fermat’s last theorem?, Math. Intellig. 16 (1994) 19-20.
• R.A. Knoebel, Exponentials reiterated, Amer. Math. Monthly 88 (1981) 235-252.

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3 Responses to “Curious Constants (part 4 of 4)”

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