Mertens Function
The summary function
where
is the Möbius Function. The first few values are 1, 0, 
, , 
, ,, , , , , , ... (Sloane's A002321). The first few values of 
at which 
are2, 39, 40, 58, 65, 93, 101, 145, 149, 150, ... (Sloane's A028442).Mertens function obeys
(Lehman 1960). The analytic form is unsolved, although Mertens Conjecture that
has been disproved.
Lehman (1960) gives an algorithm for computing 
with 
operations,while the Lagarias-Odlyzko (1987) algorithm for computing the Prime Counting Function 
can be modifiedto give in 
operations.
References
Lagarias, J. and Odlyzko, A. ``Computing Lehman, R. S. ``On Liouville's Function.'' Math. Comput. 14, 311-320, 1960. Odlyzko, A. M. and te Riele, H. J. J. ``Disproof of the Mertens Conjecture.'' J. reine angew. Math. 357, 138-160, 1985. Sloane, N. J. A. SequencesA028442 andA002321/M0102in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.: An Analytic Method.'' J. Algorithms 8, 173-191, 1987.
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