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Summatory function of the Möbius function. I: Experimental upper bounds. (Fonction sommatoire de la fonction de Möbius. I: Majorations expérimentales.) (French) Zbl 0817.11061
Let $$\mu (n)$$ be the Möbius function and $$M(x)= \sum_{n\leq x} \mu(n)$$ ($$\mu(1) =1$$, $$\mu(n) =0$$ if $$n$$ has a square divisor $$>1$$, and $$\mu(n) =1$$ or $$\mu(n) =-1$$ if $$n$$ is the product of an even or an odd number of different primes, respectively). The main interest in $$M(x)$$ stems from its connection with the Riemann hypothesis: its truth follows from the boundedness of the function $$| M(x)|/ \sqrt{x}$$. However, although it has long been thought that $$| M(x)|/ \sqrt{x}< 1$$ for $$x>1$$, it is generally believed now (but still unproved) that $$| M(x)|/ \sqrt{x}$$ tends to infinity with $$x$$.
This paper presents the results of systematic numerical computations of $$M(x)$$ for all $$x\leq 10^{12}$$, thus extending previous computations of Cohen and Dress for $$x$$ up to $$7.76\times 10^ 9$$. As a result, the upper estimate $$| M(x)| \leq 0.570591 \sqrt {x}$$, valid for $$x\in [33, 10^{12}]$$, is obtained. Moreover, a table is given of all the (eight) subintervals of $$[33, 10^{12}]$$ where $$| M(x)| >0.5 \sqrt {x}$$. Two algorithms are used for these computations: one for computing $$M(x)$$ for an isolated value of $$x$$, and a second one for computing $$M(x)$$ for many consecutive values of $$x$$. For the first algorithm it is proved that the time complexity is $$O (x^{3/4} \log^{1/2} x)$$ and the required memory is $$O (x^{1/2})$$. For the second algorithm, the time complexity is $$O( x^{1/2}\log \log x)$$ if one wants to find extremal values of $$| M(x)|/ \sqrt {x}$$ for about $$x^{1/2}$$ consecutive values near $$x$$.
Remarks. Recently, W. M. Lioen and J. van de Lune (unpublished manuscript. Dec. 1994) have extended the computations of Dress to the bound $$1.7889 \times 10^{13}$$, by using vectorized sieving; no new extrema of $$| M(x) |/ \sqrt {x}$$ were encountered.
For Part II see the following review (Zbl 0817.11062).

##### MSC:
 11Y35 Analytic computations 11Y16 Number-theoretic algorithms; complexity 11N05 Distribution of primes
Zbl 0817.11062
Full Text:
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