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Kempner Colloquium 
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\begin{Huge}\textsc{\bf $\phi$ and $\sigma$} 
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\begin{Large}\textbf{\sl Florian Luca} 
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Centro de ciencias Matem\'aticas de la Unam, Morelia, M\'exico.
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ABSTRACT


In the first part of this talk, we will survey various old and new
results related to the distribution of the values of the Euler function
$\phi(n)$ and the sum of divisors function $\sigma(n)$ of a positive integer
$n$, their popular values, their champions, as well as to the
distribution of those positive integers satisfying certain equations
involving such functions, like the perfect numbers and the amicable numbers.
In the second part of the talk, we will give some of the ideas involved in a
proof
of a recent result obtained jointly with Kevin Ford and Carl Pomerance
which says that there are infinitely common values in the ranges of these two
functions. This settles a $50$ year old question of  Paul Erd\H os.
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The talk will be accessible to a general mathematical audience.

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December 15, 2011

11:00 AM

MATH 350

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