3
$\begingroup$

How or where could I find the proof of Denjoy's probability argument for the Mertens function

$ M(x) = \sum_{n=1}^x \mu(n) = O(x^{1/2}+e) $ with $e \to 0$ based on the fact that the Möbius function $ \mu(n)$ behaves as a random variable that takes the values $\{-1,1\}$ with same probability $\frac12$?

Is there a similar probabilistic interpretation for problems inside number theory?

  • 2
    I think we're just talking about an application of the Central Limit Theorem here, aren't we? And shouldn't that $+e$ be up in the exponent? And pretty much any problem about primes can be interpreted probabilistically; numbers near $N$ are prime with probability $1/\log N$, probabilistic support for the twin primes conjecture and the Goldbach conjecture, etc., etc.2011-11-13

1 Answers 1

1

An excelent sketch of the proof for Denjoy's argument can be found in H.M. Edwards' book "Riemann's Zeta Function" page 268.