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I read about the Sigma Function today.It tells that-

The $\sigma(n)$ is the sum of all the positive divisors of $n$.

But I had no idea how they can be useful.What are the practical applications or uses of the sigma function? Are there some seemingly big mathematical problems that can be solved in seconds by applying the Sigma Function?

Thanks for any reply!!

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  • $\begingroup$ check out en.m.wikipedia.org/wiki/Divisor_function they give some good examples $\endgroup$ May 21, 2016 at 2:25
  • $\begingroup$ The Riemann hypothesis is equivalent to a inequality relating to the divisor function. $\endgroup$
    – S.C.B.
    May 21, 2016 at 2:27
  • $\begingroup$ @shaihorowitz They give examples on how to calculate the sigma values....but I am looking for applications of the sigma function.... $\endgroup$
    – Soham
    May 21, 2016 at 2:27
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    $\begingroup$ You tell me how the Mona Lisa is useful, and I'll tell you how $\sigma$ is useful. $\endgroup$ May 21, 2016 at 3:10
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    $\begingroup$ @Gerrymyerson Mona Lisa is beautiful, no use. Are you saying Riemann hypothesis is beautiful but of no use? $\endgroup$
    – N.S.JOHN
    May 21, 2016 at 5:09

3 Answers 3

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As far as I know the main properties of the sigma function are that

  1. its Dirichlet series $\zeta (s) \zeta(s-1) = \sum \sigma(n)n^{-s}$ is related to the Riemann zeta function

  2. the sum of the $\sigma$ function on intervals is the famous problem of lattice point counting in a hyperbola

I don't think there are direct applications, but 1-2 provide a relation between some difficult counting problems in analytic number theory and the easier problem of sums over lattice points.

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a general relationship between how n's prime factorization effects (n+1) prime factorization would offer a proof to the reiman hypothesis, collatz conjecture, and more conjectures then I can count.

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  • $\begingroup$ I don't think I see how this is related to the "sum of divisors" function. Could you elaborate? $\endgroup$
    – pjs36
    May 21, 2016 at 2:54
  • $\begingroup$ consider sigma (p) p prime well then it's divisors are 1 and P so we have 1+p as divisors now consider sigma (p+1) what do we know? that's at the heart of this function $\endgroup$ May 21, 2016 at 2:57
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The sigma function also shows up naturally in the study of modular forms and elliptic curves, which makes it significant in algebraic geometry (hence physics) and in cryptography.

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