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As we can see in the plot below, Mertens function: $M(x)\equiv \sum_{n=1}^{x}\mu(n)$ has wild swings from positive to negative and back again.

merten's plot

When we use: $$x=\frac{1}{2+\frac{1}{\phi+2}}\text{, where }\phi \text{ is the golden ratio,}$$ as the power for each $n$ we get the blue line and using its negative we get the red line. Both lines hug the extremes of the sum quite nicely. I have checked to about $10^{10}$ without finding any exceptions.

I have found only one reference to $\phi+2$ in the literature, specifically in "Mathematical Constants" by Steven R. Finch, page 418, where it is shown as the $\text{Tutte-Beraha constant} B_{10}$.

My question is: could $\phi+2$ be used to explain the behavior of the Mertens function? Or would this be a case of the Law of Small Numbers?

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The name of the mathematician behind this is Mertens; I thus changed Merten's to Mertens; if you would still like to have some possessive s please reintroduce it in a consistent way. –  quid Mar 30 '13 at 16:13
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2 Answers

up vote 10 down vote accepted

I'm not entirely sure I understand the question. However, I believe you are asking if the inequality $|M(x)| \leq x^{\frac{1}{2+1/(\phi+2)}}$ might hold.

It is known that the Mertens function satisfies $M(x) \geq 1.2 \sqrt{x}$ (as well as $M(x) \leq - 1.2 \sqrt{x}$) infinitely often (see the paper "The Mertens Conjecture Revisited" by Tadej Kotnik and Herman te Riele, although this is classical with a constant smaller than 1).

This implies that your inequality is violated infinitely often for larger values of $x$.

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While the specific question appears to be answered, I would like to add a more general one:

The study of the Mertens function $M(x)= \sum_{n \le x} \mu(n)$ is a notorius problem, and (thus) any new insight based on numerical investigations for very small values (in this context) seems most unlikely.

The function $M(x)$ is in a very vague sense about square-root-ish and thus one sometimes writes $q(x)=M(x)/\sqrt{x}$.

On the one hand, there used to be an old conjecture that $|q(x)|\lt 1$ (Mertens' conjecture), which was refuted by Odlyzko and te Riele (1985), and was already considered very unlikely to be true before, and Mark Lewko mentioned the curent 'record' constants. But it is beleived that $q(x)$ is in fact unbounded.

On the other hand, an estimate $q(x)= O(x^{\varepsilon})$ for every $\varepsilon > 0$ is equivalent to the Riemann Hypothesis. More precisely, by a recent result of Soundararajan it is known that under RH one has $$ q(x)= O(\exp( \sqrt{\log x} (\log \log x)^{14})), $$ and Balazard and de Roton showed that $14$ can be optimized to $5/2 + \varepsilon$ for every $\varepsilon > 0$.

Yet, it seems to be not clear (even conjecturally) how $q(x)$ should actually behave.

Kotnik and van de Lune (Exp. Math. 13.4) made the conjecture that $$ q(x)= \Omega_{\pm}(\sqrt{ \log \log \log x}), $$ and Kotnik and te Riele (mentioned in Mark Lewko's answer) discuss that extremal observed values of $q(x)$ are close to $\pm \frac{1}{2}(\sqrt{ \log \log \log x})$.

However, and as mentioned there, if this would remain (about) true 'forever' this would contradict Ng (2004) [and Gonek (unpublished)] conjecture that limes superior and limes inferior of $$ \frac{q(x)}{( \log \log \log x)^{5/4}} $$ are in fact $\pm B$ for some positive and finite $B$.

Yet, in the in the 70's still other conjectures were made namely that the limit of $$ \frac{|q(x)|}{\sqrt{ \log \log x}} $$ should exist, and even two(!) values were suggested for the limit. [Note: there are only two log's here.]

And, there would still be different contributions to this. For example, Kaczorowski (Journal London Math. Soc. 2007) showed that a 'twisted' version of $M(x)$ is fairly large, namely $$ \sum_{n \le x} \mu(n) (\cos (x/n) -1) = \Omega_{\pm}(\sqrt{x} \log \log \log x) $$ and he derives from this that for every real $a\neq 0$ $$ |\sum_{n \le x} \mu(n)|+|\sum_{n \le x} \mu(n) \cos (ax/n)| = \Omega(\sqrt{x} \log \log \log x) $$ which would, if one could take $a=0$, imply $|q(x)| = \Omega (\log \log \log x)$. Or, put differently, shows that if the $|q(x)|$ is not as large, the sum with the cosine has to be large for every non-zero $a$. In contrast to the idea that $\frac{1}{2}(\sqrt{ \log \log \log x})$ might be about right.

In any case, this problem is complicated in that even detailed and recent investigations can arrive at different conclusions what should or might be the right expectation regarding the behavior of $M(x)$.

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