Small values of the Euler function and the Riemann hypothesis

Details Small values of the Euler function and the Riemann hypothesis Small values of the Euler function and the Riemann hypothesis

Publishing date

2012-02-03

URL

arxiv.org/abs/1202.0729v2

Science

Mathematics

Field

Number Theory

Type

Scientific paper

Abstract

Let $\vfi$ be Euler's function, $\ga$ be Euler's constant and $N_k$ be the
product of the first $k$ primes. In this article, we consider the function
$c(n) =(n/\vfi(n)-e^\ga\log\log n)\sqrt{\log n}$. Under Riemann's hypothesis,
it is proved that $c(N_k)$ is bounded and explicit bounds are given while, if
Riemann's hypothesis fails, $c(N_k)$ is not bounded above or below.

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