[Fwd: Max value of x/phi(x)]

Pieter Moree moree at science.uva.nl
Mon Sep 13 22:28:57 CEST 2004

This is well-known in elementary number theorem since Landau (1909).
lim inf x/\phi(x)=1 (taking x to be primes)
lim sup x/(\phi(x) log log x)=e^{gamma},
with gamma Euler's constant (taking x to be products of consecutive
primes starting from 2 and applying Mertens' theorem).

For more results in this direction vide e.g.
The new book of Prime Number Records of P. Ribenboim, pp. 319-320.

Pieter Moree

>Does anyone know what the maximum possible value of x/phi(x) can be
(where >phi is Euler's totient function)? Is there a theorem in regards
to this?

>From some playing around with Mathematica it seems to me that the
value can be made as large as one wants by choosing a sufficiently large
highly composite number, but I'm wondering if x/phi(x) is
bounded by some property of x, such as its square root.

Alonso Delarte

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