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I would like gain detailed knowledge about properties of the divisor sum function $\sigma(n)$, special equation that have been studied (e.g. $\sigma(n) = 2n$ perfect numbers, ...) and progress that was made in the past centuries.

I am interested in arithmetic functions in general so the books mentioned in this question Good books on Arithmetic Functions ? is already a good start.

However, it would be great if you could recommend a book or a comprehensive paper that introduces/summarizes knowledge about the $\sigma(n)$ function.

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For me a very good source for those matters, comprehensive in many aspects and adding enlightening notes and exercises, is

Tenenbaum G., Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1995

It has just been reedited by the AMS. Moreover, he also wrote a book Divisors, at Cambridge Tracts. Despite I don't know what is inside, it would be worth a look.

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Sándor, Jó.; Mitrinović, D. S. & Crstici, B. Handbook of number theory. I Springer, 2006

This book describes a lot of results concerning almost all arithmetical functions.

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Hint: try to check this online book under this title The math encyclopedia of smarandach type notion in number theory include also arithmitic function , And bellow papers , However they include iterate sum of divisor function, but you w'ill find solution of your problem , papers include " meta perfect number:"

Cohen, Graeme L., and Herman JJ te Riele. "Iterating the sum-of-divisors function."

Kátai, I. "On the prime power divisors of the iterates of $\phi(n)$ and $\sigma(n)$, Šiauliai

Math." Semin 4.12 (2009): 125-143 Experimental Mathematics 5.2 (1996): 91-100.

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