On the order of the Mertens function. (English) Zbl 1065.11078

Kotnik, Tadej; van de Lune, Jan

Exp. Math. 13, No. 4, 473-481 (2004).

The authors give an interesting account of numerical experiments concerning \[ q(x)={1\over\sqrt x}\sum_{n\leq x}\mu(n), \] where \(\mu(n)\) is the Möbius function. Since \(q(x)\) behaves atypically for small \(x\), in what follows the range concerned is \(10^4\leq x\leq 10^{10^{10}}\), and only ‘increasingly large extrema for \(q\)’ are considered.
For \(x\) with modest size, the value for \(q(x)\) can be computed accurately, and there are seven increasingly large \(q\) values in the range \(10^4\leq x\leq10^{14}\). The first negative one is \(q(24184)=-0{\cdot}462{\ldots}\,\), the fourth one is \(q(71578936427177)=-0{\cdot}524{\ldots}\,\), and the first positive one is \(q(48433)=+0{\cdot}436{\ldots}\,\), the third one is \(q(7766842813)=+0{\cdot}570{\ldots}\,\).
For larger values of \(x\), an approximation \(q_K(x)\) with a parameter \(K\) is computed instead; the definition for \(q_K(x)\) is too involved to be given here, but \(K\) is set as \(10^2,10^4\) and \(10^6\) in the experiments. Naturally care has to be taken to ensure that the approximations being computed do relate to the extreme values being sought after, and although the procedure becomes questionable if one is trying gain insight for an \(O\)-estimate, it is sound when applied to an \(\Omega\)-estimate. An algorithm is given for the delivery of increasingly large values for \(q_K\) in the range \(10^{20}\leq x\leq 10^{10000}\), and a more complicated procedure is involved in the search for the range \(10^{10000}< x\leq 10^{10^{10}}\).
The accompanying figures for extreme values for \(q_K\) show that they lie near \(\pm{1\over2}\sqrt{\log\log\log x}\), although the authors are only prepared to conjecture that \(q(x)=\Omega_{\pm}(\sqrt{\log\log\log x}\,)\). Nevertheless, the conjecture is already in conflict with other conjectures about the order of \(q(x)\), and some discussions are offered concerning whether there is randomness or regularity in the distributions of positive and negative values of \(\mu(n)\).

Reviewer: Peter Shiu (Loughborough)

MSC:

11N56	Rate of growth of arithmetic functions
11Y35	Analytic computations
11-04	Software, source code, etc. for problems pertaining to number theory

References:

[1]	Deléglise M., Exp. Math. 5 pp 291– (1996)
[2]	Dress F., Exp. Math. 2 pp 93– (1993)
[3]	DOI: 10.1112/S0024611502013655 · Zbl 1034.11044 · doi:10.1112/S0024611502013655
[4]	Good I. J., Math. Comp. 22 pp 857– (1968)
[5]	Kotnik, T. and van de Lune, J. 2003.”Further Systematic Computations on the Summatory Function of the Möbius Function.”1–9. [Kotnik and van de Lune 03],CWI ReportMAS-R0313
[6]	DOI: 10.1016/0196-6774(87)90037-X · Zbl 0622.10027 · doi:10.1016/0196-6774(87)90037-X
[7]	Littlewood J. E., C. R. Acad. Sci. 158 pp 1869– (1917)
[8]	Mertens F., Sitzungsber. Akad. Wiss. Wien IIa 106 pp 761– (1897)
[9]	DOI: 10.1112/S0024611504014741 · Zbl 1138.11341 · doi:10.1112/S0024611504014741
[10]	Odlyzko A. M., J. Reine Angew. Math. 357 pp 138– (1985)
[11]	Pintz J., Astérisque 147 pp 325– (1987)
[12]	Saffari B., C. R. Acad. Sci. A 271 pp 1097– (1970)
[13]	te Riele H. J. J., Thomas Jan Stieltjes–Collected Papers 1 pp 69– (1993)
[14]	DOI: 10.1112/jlms/s1-2.4.247 · JFM 53.0312.02 · doi:10.1112/jlms/s1-2.4.247
[15]	Titchmarsh, E. C. 1951.The Theory of the Riemann Zeta-function318–325. Oxford, UK: Oxford University Press. [Titchmarsh 51]
[16]	von Mangoldt H., J. Reine Angew. Math. 114 pp 255– (1895)