# Asymptotic expansion for Mertens function

1. Feb 20, 2008

### mhill

My conjecture is that using the Selberg-Delange method the next asymptotic equality holds

$$\sum_{n \le N } \mu (n) \sim (\sqrt n )log^{a}(n) \frac{G(c)}{\Gamma (b)}$$

where a,b and c are positive constants and Gamma stands for (n-1)! , G(c) is just $$G(c) \zeta (c) =1$$

2. Feb 20, 2008

### CRGreathouse

I think it's quite unlikely that the Mertens function settles down into an asymptotic equality, eljose! Hasn't some kind of result been proved to the effect that it crosses zero an infinite number of times?

Further, I think the conjecture implies the Riemann hypothesis, since it would mean that M(n) is bounded by $$o(n^{1/2+\varepsilon})$$ -- right?