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)
The first explicit estimate for 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 for any D0. 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)
Acta Arith., 13. Erratum, ibid. 16 (1969), 99-100.
].
Theorem (1967)
When D0, we have |M(D)|180D+5. When D1119, we have |M(D)|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) and for the Möbius function M(X)
Acta Arith., 52, 307--337.
] shows that
Theorem (1993)
When D120727, we have |M(D)|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, we have |M(D)|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, we have |M(D)|0.571D.
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-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}-function
Funct. Approx. Comment. Math., 37(, part 1), 51--63.
].
Theorem (1996)
When D2160535, we have |M(D)|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, [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, we have |M(D)|/D2.9/logD.
[El Marraki, 1995 †El Marraki, M. 1995
Fonction sommatoire de la fonction μ de Möbius, majorations asymptotiques effectives fortes
J. Théor. Nombres Bordx., 7(2).
] improves that into
Theorem (1995)
When D685, we have |M(D)|/D0.10917/logD.
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, we have |M(D)|/D0.013/logD.
2. Bounds on m(D)=dDμ(d)/d
[Mac Leod, 1967 †Mac Leod, R.A. 1967
A new estimate for the sum M(x)=nxμ(n)
Acta Arith., 13. Erratum, ibid. 16 (1969), 99-100.
] shows that the sum m(D) takes its minimal value at 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.
] and reads
Theorem (1996)
When D0 and for any integer r1, we have dD,(d,r)=1μ(d)/d1.
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 and for any integer r1 and any real number ε0, we have dD,(d,r)=1μ(d)/d1+ε1+ε.
Concerning upper bounds that tend to 0, [El Marraki, 1996 †El Marraki, M. 1996
Majorations de la fonction sommatoire de la fonction μ(n)n
Univ. Bordeaux 1, Pré-publication(96-8).
] is the first to get such an estimate.
Theorem (1996)
When D33 we have |m(D)|0.2185/logD.
When D>1 we have |m(D)|726/(logD)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 we have |m(D)|0.026/logD.
[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

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