“Mertens Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Mertens Function

释义

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.

See also Mertens Conjecture, Möbius Function

References

Lagarias, J. and Odlyzko, A. ``Computing : An Analytic Method.'' J. Algorithms 8, 173-191, 1987.

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.

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