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Bulletin of the London Mathematical Society Advance Access originally published online on March 6, 2009
Bulletin of the London Mathematical Society 2009 41(2):213-226; doi:10.1112/blms/bdn119
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© 2009 London Mathematical Society

The sum of the Möbius function

H. Maier

Abteilung Zahlentheorie und Wahrscheinlichkeitstheorie
Helmholtzstrasse 18
Universität Ulm
89069 Ulm
Germany
helmut.maier@uni-ulm.de

H. L. Montgomery

Department of Mathematics
University of Michigan
Ann Arbor, MI 48109-1043
USA

Received 10 September 2007. Revision received 31 January 2008.

We derive from the Riemann Hypothesis an estimate for M(x) = {sum}n≤xµ(n). This is the first improvement of the bound that Titchmarsh established in 1927.

2000 Mathematics Subject Classification 11N56, 11M26.

Research supported in part by NSF FRG Grant DMS-0244660 and NSF Grant DMS-0653529.


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