Explicit bounds on the Moebius function

Collecting references: [Diamond & Erdös, 1980 †Diamond, H.G., & Erdös, P. 1980
On sharp elementary prime number estimates
Enseign. Math., 26(3-4), 313--321.
], [Deléglise & Rivat, 1996b †Deléglise, M., & Rivat, J. 1996b
Computing the summation of the Möbius function
Exp. Math., 5(4), 291--295.
], [Borwein et al., 2008 †Borwein, P., Ferguson, R., & Mossinghoff, M.J. 2008
Sign changes in sums of the Liouville function
Math. Comp., 77(263), 1681--1694.
].
1. Bounds on M(D)=dDμ(d)$M(D)=\sum_{d\le D}\mu(d)$
The first explicit estimate for M(D)$M(D)$ is due to [von Sterneck, 1898 †von Sterneck, R.D. 1898
Bemerkung über die Summierung einiger zahlentheorischen Funktionen
Monatsh. Math. Phys., 9, 43--45.
] where the author proved that |M(D)|19D+8$|M(D)|\le \tfrac19 D+8$ for any D0$D\ge0$. A popular estimate is the one of [Mac Leod, 1967 †Mac Leod, R.A. 1967
A new estimate for the sum M(x)=nxμ(n)$M(x)=\sum_{n\le x}\mu(n)$
Acta Arith., 13. Erratum, ibid. 16 (1969), 99-100.
].
Theorem (1967)
When D0$D\ge 0$, we have |M(D)|180D+5$|M(D)|\le \tfrac1{80} D+5$. When D1119$D\ge 1119$, we have |M(D)|D/80$|M(D)|\le D/80$.
We mention at this level the annoted bibliography contained at the end of [Dress, 1983/84 †Dress, F. 1983/84
Théorèmes d'oscillations et fonction de Möbius
Sémin. Théor. Nombres, Univ. Bordeaux I, Exp. No 33, 33pp. http://resolver.sub.uni-goettingen.de/purl?GDZPPN002545454.
]. [Costa Pereira, 1989 †Costa Pereira, N. 1989
Elementary estimates for the Chebyshev function ψ(X)$\psi(X)$ and for the Möbius function M(X)$M(X)$
Acta Arith., 52, 307--337.
] shows that
Theorem (1993)
When D120727$D\ge 120\,727$, we have |M(D)|D/1036$|M(D)|\le D/1036$.
On elaborating on this method, [Dress & El Marraki, 1993 †Dress, F., & El Marraki, M. 1993
Fonction sommatoire de la fonction de Möbius 2. Majorations asymptotiques élémentaires
Exp. Math., 2(2).
] showed that
Theorem (1993)
When D617973$D\ge 617\,973$, we have |M(D)|D/2360$|M(D)|\le D/2360$.
One of the argument is the estimate from [Dress, 1993 †Dress, F. 1993
Fonction sommatoire de la fonction de Möbius 1. Majorations expérimentales
Exp. Math., 2(2).
]
Theorem (1993)
When 33D1012$33\le D\le 10^{12}$, we have |M(D)|0.571D$|M(D)|\le 0.571\sqrt{D}$.
Another tool is [Cohen & Dress, 1988 †Cohen, H., & Dress, F. 1988
Estimations numériques du reste de la fonction sommatoire relative aux entiers sans facteur carré
Prépublications mathématiques d'Orsay : Colloque de théorie analytique des nombres, Marseille, 73--76.
] where refined explicit estimates for the remainder term of the counting functions of the squarefree numbers in intervals are obtained.
The latest best estimate of this shape comes from [Cohen et al., 1996 †Cohen, H., Dress, F., & El Marraki, M. 1996
Explicit estimates for summatory functions linked to the Möbius μ$\mu$-function
Univ. Bordeaux 1, Pré-publication(96-7).
]. This preprint being difficult to get, it has been republished in [Cohen et al., 2007 †Cohen, H., Dress, F., & El Marraki, M. 2007
Explicit estimates for summatory functions linked to the Möbius {μ$\mu$}-function
Funct. Approx. Comment. Math., 37(, part 1), 51--63.
].
Theorem (1996)
When D2160535$D\ge 2\,160\,535$, we have |M(D)|D/4345$|M(D)|\le D/4345$.
These results are used in [Dress, 1999 †Dress, F. 1999
Discrépance des suites de Farey
J. Théor. Nombres Bordx., 11(2), 345--367.
] to study the discrepancy of the Farey series.

Concerning upper bounds that tend to 0$0$, [Schoenfeld, 1969 †Schoenfeld, L. 1969
An improved estimate for the summatory function of the Möbius function
Acta Arith., 15, 223--233.
] is the pioneer and shows among other estimates that
Theorem (1969)
When D>0$D>0$, we have |M(D)|/D2.9/logD$|M(D)|/D\le 2.9/\log D$.
[El Marraki, 1995 †El Marraki, M. 1995
Fonction sommatoire de la fonction μ$\mu$ de Möbius, majorations asymptotiques effectives fortes
J. Théor. Nombres Bordx., 7(2).
] improves that into
Theorem (1995)
When D685$D\ge 685$, we have |M(D)|/D0.10917/logD$|M(D)|/D\le 0.10917/\log D$.
The latest bound coming from [Ramaré, 2013a †Ramaré, O. 2013a
From explicit estimates for the primes to explicit estimates for the Moebius function
Acta Arith., 157(4), 365--379.
] improves that:
Theorem (2012)
When D1100000$D\ge 1\,100\,000$, we have |M(D)|/D0.013/logD$|M(D)|/D\le 0.013/\log D$.
2. Bounds on m(D)=dDμ(d)/d$m(D)=\sum_{d\le D}\mu(d)/d$
[Mac Leod, 1967 †Mac Leod, R.A. 1967
A new estimate for the sum M(x)=nxμ(n)$M(x)=\sum_{n\le x}\mu(n)$
Acta Arith., 13. Erratum, ibid. 16 (1969), 99-100.
] shows that the sum m(D)$m(D)$ takes its minimal value at D=13$D=13$. A folklore result is generalized in [Granville & Ramaré, 1996 †Granville, A., & Ramaré, O. 1996
Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients
Mathematika, 43(1), 73--107.
Theorem (1996)
When D0$D\ge 0$ and for any integer r1$r\ge1$, we have dD,(d,r)=1μ(d)/d1$\Bigl|\sum_{\substack{d\le D,\\ (d,r)=1}}\mu(d)/d\Bigr|\le 1$.
In fact, Lemma 1 of [Davenport, 1937 †Davenport, H. 1937
On some infinite series involving arithmetical functions
Quart. J. Math., Oxf. Ser., 8, 8--13.
] already contains the requisite material. This is further extended in [Ramaré, 2013b †Ramaré, O. 2013b
Some elementary explicit bounds for two mollifications of the Moebius function
Functiones et Approximatio, 229--240.
] where it is shown that
Theorem (2012)
When D0$D\ge 0$ and for any integer r1$r\ge1$ and any real number ε0$\varepsilon\ge0$, we have dD,(d,r)=1μ(d)/d1+ε1+ε$\Bigl|\sum_{\substack{d\le D,\\ (d,r)=1}}\mu(d)/d^{1+\varepsilon}\Bigr|\le 1+\varepsilon$.
Concerning upper bounds that tend to 0$0$, [El Marraki, 1996 †El Marraki, M. 1996
Majorations de la fonction sommatoire de la fonction μ(n)n$\frac{\mu(n)}n$
Univ. Bordeaux 1, Pré-publication(96-8).
] is the first to get such an estimate.
Theorem (1996)
When D33$D\ge33$ we have |m(D)|0.2185/logD$|m(D)|\le 0.2185/\log D$.
When D>1$D>1$ we have |m(D)|726/(logD)2$|m(D)|\le 726/(\log D)^2$.
[Ramaré, 2013a †Ramaré, O. 2013a
From explicit estimates for the primes to explicit estimates for the Moebius function
Acta Arith., 157(4), 365--379.
] proves several bounds including the following one.
Theorem (2012)
When D61000$D\ge 61\,000$ we have |m(D)|0.026/logD$|m(D)|\le 0.026/\log D$.
[Balazard, 2012 †Balazard, M. 2012
Elementary Remarks on Möbius' Function
Proceedings of the Steklov Intitute of Mathematics, 276.
].
3. Bounds on m~(D)=dDμ(d)log(D/d)/d$\tilde{m}(D)=\sum_{d\le D}\mu(d)\log(D/d)/d$

Last updated on July 20th, 2012, by Olivier Ramaré