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OEIS (A028442) lists the Numbers n such that Mertens' function
$$ M(n)=\sum_{k=1}^n\mu(k) $$ is zero:

2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427,...

Do these numbers have a deeper significance other than: The set of numbers below $n$ is split into $2$ equally large sets with $\mu(m\le n)=\pm 1$ (with asymptotic density each $\frac{3}{\pi^2}$) and the set $\mu(m\le n)=0$ (with asymptotic density $1-\frac{6}{\pi^2}$)?

I mean, does the fact that $\lim_{n\to\infty}M(n)=0$ play a role (whatever that is in the infinite case) in a finite case as well? I'm especially interested in the case where $n$ is even.

draks ...
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  • See also http://math.stackexchange.com/questions/213260/approximate-how-the-numbers-n-such-that-mertens-function-is-zero-grow. – joriki Oct 13 '12 at 20:25
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    It's not a fact that $M(n) \to 0$, obviously (since $\mu(n) = 1$ infinitely often). – Erick Wong Jan 08 '13 at 08:36
  • @ErickWong so it's not sufficient that the asymptotic density equal each other? And why should there be more numbers $n$ with an even number of factors $(\mu(n_e)=1)$ compared to an odd number of factors$(\mu(n_o)=-1)$? – draks ... Jan 08 '13 at 11:49
  • @draks... You're confusing $M(n) \to 0$ with $M(n) = o(n)$. The latter is true (and somewhat deep, being equivalent to PNT), the former is certainly false (by the most trivial test of divergence). – Erick Wong Jan 08 '13 at 17:37
  • @ErickWong I found this one: ... while the last one is obvious if we allow generalized summation $\sum_{n=1}^{\infty}\mu(n) = \frac{1}{\zeta(0)} = -2$ (see here). What do you think? – draks ... Jan 10 '13 at 21:36
  • @draks... Not sure where you're going with this. This is some generalized summation (perhaps through analytic continuation of Dirichlet series?). Seems to me that your question is about standard summation, since it concerns the partial sums of $\mu(n)$. – Erick Wong Jan 11 '13 at 23:35
  • Not sure if this helps...http://math.stackexchange.com/q/314383/28555 – Fred Daniel Kline Feb 27 '13 at 12:18
  • @Fred how do you think it could? – draks ... Jul 22 '18 at 18:19
  • @draks..., see https://oeis.org/A019565 – Fred Daniel Kline Jul 22 '18 at 23:15

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